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Saturday, June 25th 2016


The marvels go on and on.


No. 14 Tractor

Eight lanes back and forth at the pool Thursday. Only four on Friday, but I was pretty exhausted before I ever got into the water. On Thursday I had taken the red No. 14 Tony Stewart and A.J. Foyt Sears Craftsman Silberpfeil tractor out to cut the grass and wound up with a flat rear tire. Friday, I tried to fix it, but each stage—removing the wheel, jacking up the tractor, pumping air into the tire, etc., not necessarily in that order—came with its own sub-problems to the point where I finally decided that on Monday I’ll take the wheel to Troy Greer’s and ask them to, please, fix it. I'm hoping that they won't just inform me that the rim is bent. Today (Saturday) the pool was too crowded to do a lot of uninterrupted laps.

In line with my normal practice, I’m not making a whole lot of progress on this series as a whole because I keep running across information that I find to be either necessary, helpful, or just plain cool. So, this post also does not go nearly as far as I wanted it to, but at least I have started to raise the theological questions.

The ubiquity of the Fibonacci numbers and their steady companion, ϕ, has become the occasion for much thought and writing about the mathematical regularity and apparent intentionality of the universe. In fact, the closer we look, the more astounding things become.

Again, because in this series I’m trying to stress the beauty of the numbers themselves, let me give you another example that strikes me as really astounding. We all know about ϕ’s big brother pi (π) mostly for his part in the formula for the area of a circle:

A =πr2

But π also shows up in places where we would not expect to see him. Skip down if you like.

Jakob BernoulliLet us look at an interesting problem, known as the "Basel Problem" (Derbyshire 63) because it was in Basel, Switzerland, that it was posed. The problem, as originally stated by Jakob Bernoulli (1655-1705), went as follows.

Here is a convergent series, which means that it eventually approaches the limit of a specific number rather than diverging to infinity:

Basel Problem

As usual, we can come closer and closer to the number, but will never quite reach it. A precise enough approximation of the number is

1.64493406693 …

Are you getting tired of those three little dots ( ), which indicate that you’ll never truly get to the end of calculating the formula’s value? Apparently Bernoulli was. In a publication on various related topics, he included this series and asked for input from other mathematicians about this question. Would anyone be able to state it in a form that did not lose itself in the forever of infinity, but in a “closed” formula that could be substituted for it?

Leonhard Euler (1707-1783), whose original home also happened to have been Basel, came up with a solution:

Basel Soution

I don't think I'd be going out on a limb when I say that most of us are rather surprised to see pi popping up there.

Very quickly let's switch to a different topic, this time probability theory. Suppose you are asked to pick two numbers at random from a set of integers. What is the probability that they will not have any common divisors? E.g., 6 and 8 would share a divisor (2), but 4 and 9 would not. The probability of picking two numbers without a common divisor is:


The actual number is ~0.6079 or ~60.79%, but that’s obviously not what strikes us. This probability calculation turns out to be the reciprocal of Euler's solution to the Basel problem! The University of Illinois has set up an interactive webpage that illustrates this formula with some input from you.

The marvels go on and on.

Ben Stein Expelled

Christians, Jews, and others who believe in a personal God look at these wonders of the universe and see in them the magnificent hand of its Creator. How could they not? I’m not offering that sentiment as a piece of apologetic on behalf of the created-ness of the world or the existence of God, but as a report on what is (or should be) an obvious part of what theists believe. Or at least should. Ben Stein, whom I was privileged to count among my blog readers for a while, made the film “Expelled: No Intelligence Allowed” to publicize the fact that the scientific world on the whole (including atheists and purported Christians) has slammed the door shut on even asking the question of whether the cosmos manifests intelligent design. The entire movie is available on YouTube There are exceptions, such as William Dembski, who has not allowed his views to be dictated by the strong arm of a supposed scientific consensus, at the expense of his ongoing career. As Mr. Stein's movie displays, even atheists who allow the question to be raised may find their jobs terminated.

There is another group of scientists, however, including some of the brightest people of the last 100 years, who have not been afraid to express belief in God on the basis of the mathematical coherence of the universe, although they do not subscribe to a traditional understanding of God. The concept of God, as they have been espousing it, is not that of a personal Creator, and it doesn’t even really fit into any other traditional categories. It is not pantheism in which God and the universe are considered to be identical; let alone deism according to which God created the universe and then abandoned it to run according to the program he implanted in it. In some ways it’s reminiscent of the God of process theology who is persuading the world to follow his directives, from the cohesion of molecules on up. Still, there is far too much metaphysics associated with process theism to suit these celebrity believers. It appears to be the very regularity of the universe, the math behind the phenomena, so to speak, that constitutes God for them. Next time I’ll continue with these observations on Albert Einstein and Michio Kaku, and then return to phi.

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Wednesday, June 22nd 2016


Excursus: A Corn Stack


Woops! I meant to go on talking about ϕ, but got caught up in the agriculture of Indiana by way of a remark on the weather. That's why I like to describe my blog as "mercurial." Long-time readers will know that there have been other times when I was surprised at what I wrote about rather than what I intended to write about.

For the last few days I had to go without a certain medication due to some snafus beyond my control, and I’m very glad to have it back. It was not good, and that's all I can--or maybe want to--say about that.

A couple of cloudy, rainy days, but the temperatures still are definitely in the summer range. If you should drive, bike, or walk to the outskirts of Smalltown, USA, you will find yourself facing corn fields, and you might just hear the corn growing. The standard expectation is that the corn should be “knee high by the Fourth of July.” Many of the fields are knee high already, and will be close to “person high” by Independence Day, I should think. The fields definitely will dominate the view for the next few months.

It looks to me as though the timing worked out quite well for planting corn (aka maize, btw) this year. The big question is always when the fields will be dry enough for tractors and machinery not to sink in the mud. Around March or April, after the snow has melted and we’ve gotten our usual overdose of rain, many of the fields look like lakes. The farmer has to wait for the soil to dry out, and appearances can be deceiving in that regard. An early unusually warm spell can make the fields look dry, but that part may only be a crust with a lot of mud underneath. This is not helpful, for one thing because the farmer’s tractor might get stuck, and for another because such a crust also stands in the way of further evaporation of the water underneath the crust.

Around here, for corn to have enough time to reach full maturity, it needs to be “in” by “Decoration Day,” as I once overheard a farmer say.

  • Excursion: That’s “Memorial Day” for most of us. Among the rural folks here whose social life centers on the “Post” or the “Legion,” it is indeed still “Decoration Day.” Ruby, a sprightly 94-year-old (at the time) lady at one of the churches where I used to preach, would go to the local cemetery first thing in the morning of Memorial Day and make sure there was an American flag planted on the grave of each former service man or woman. Furthermore, just in case she had accidentally omitted someone, she would place two crossed flags at the edge of the cemetery so that they would be included as well. – I find these kinds of observances intriguing, particularly in light of their analogues in other cultures.

Anyway, I’m pretty sure that everyone got their corn planted before Decoration Day. One must realize, of course, that there are different kinds of corn with different requirements. Overwhelmingly, the corn grown here is “horse corn” earmarked to become animal feed or silage. It lacks the appeal of the traditional “sweet corn” which is a different species altogether. Horse corn stays out in the field until it's all dried out. You wouldn't want to eat an ear of sweet corn in that condition.

Van Buren, Indiana Speaking of corn, let’s say that you want to watch a movie, and you think a bowl of popcorn would really go well with it. Great! Just be sure you have real popping corn. If you just take a handful of any old type of corn, put it in your microwave or other appliance, and stand by, ready with salt and butter, you may find that it doesn’t work that way. The kernels are not going to spring to life, and you can put the salt and butter back in the cupboard. Popcorn is a different kind of corn from the others mentioned so far. It is distinguished by the fact that, when the kernels have matured, they surround a tiny little drop of water. The heat brings that little bit of moisture to the boiling point, which increases the internal pressure, and the kernel, along with most of its colleagues in the bag, explodes into the white treat we’re all familiar with. So, popcorn is also among the agricultural items that farmers plant around here. About half an hour’s drive from here there is the little town of Van Buren, Indiana, population 864 as of 2010 (Compare to Smalltown, USA,’s population of 5,145. Both are shrinking.) It declares itself to be the Popcorn Capital of the World and celebrates an annual popcorn festival.

But, as another saying goes, “There’s more than corn in Indiana.” Beans (i.e. soy beans, though nobody call them that around here), go in about a week or two later than corn. There are also a few occasional wheat fields; it’s “winter wheat,” which means that it was planted way back in last October or so. You don’t see anything of it all winter, but these fields become some of the earliest and brightest displays of green when spring arrives. By now the wheat has attained its golden color, and it should be ready for harvesting pretty soon. The farmers are also getting ready for the first round of mowing hay.

As long as I’m pursuing this subject, you will also see occasional tomato fields in Indiana and surrounding states. According to the Wikipedia's statistics, if you’re in Indiana and you see a tomato field, there’s a 95% chance that the harvest will be processed into one of the many kinds of canned tomato products at the Red Gold plant, just a short three miles west of here in tiny Orestes, Indiana (pop. 414, not shrinking). Red Gold also has contracts with 80% of the tomato growers in the rest of the Midwest.

  • Excursus: Red Gold still likes to present itself as a mom & pop organization, and maybe it is. The daughter or granddaughter of the owners, I don’t remember which, was in Seth’s 4-H chapter, and went along on at least one fossil and mineral hunting trip with us. By “us” I mean a few 4-H’ers, the leader (Archie H.) who knew all of the best quarries in the state (and all of their managers, and all of their children, and all of their names), and yours truly driving. Still, when I tell you that in 1993 “mom and pop” acquired Stokely-Van Camps’ tomato division, you may see that they’re a major player.

I might mention that, despite a strong heritage of sorghum (aka milo) production in Indiana, it is no longer a major crop here. The Hoosier State had been the nation’s leader in this regard at one time, but it became unprofitable. Nevertheless, there still are a number of sorghum fields, even if the US department of agriculture doesn’t acknowledge their existence. In its early stages, corn and sorghum look a lot alike; the distinction being that sorghum plants are a little shorter than corn and its leaves are spikier. Later on, though, sorghum clearly stands out with much larger and wider tassels than you’ll see on corn.


Pictures courtesy of Sorghum: The Smart Choice--All About Sorghum


Some efforts are being made to reintroduce sorghum as a profitable crop on both public and private levels.

Next time, if I can keep my mind from running off into different directions, the perennial question: Is God a Mathematician?

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Monday, June 20th 2016


Bull's Eye (almost)


Most of this was written originally on Saturday evening, but needed some refinement and a lot of uploading of formulas, plus adding enough cute stuff to give it appeal to a general public. Hope you like the Staypuft Marshmallow man gif I concocted.

The weather continuous to be quite nice, and definitely appropriate for summer. We are heading into the high nineties again today. A couple of weeks ago I procured my annual season pass to the Beulah Park Pool, here in Smalltown, USA, and I’m making good use of it, every day if I at all possible. It’s funny (so to speak): on the first day I went out there this year, I felt like I had lost an enormous amount of strength in my arms and legs, and I wasn’t surprised. I couldn’t even manage to swim one entire lane without taking a break. However, by this past Friday, after only about a week, I managed four, and as of yesterday and today, I’m up to six already. That’s definitely also been a surprise, and obviously a good one, for which I’m grateful. On most days, after swimming one set, I usually take a break and visit the water slide once or twice to satisfy my need for speed. Then I do another set, casually and without trying to go to any limit.

Well, let’s get back to the numbers.

StayPuff Marshmallow Man

There are several mathematical surprises connected to the Fibonacci series in its own right, apart from its involvement with ϕ. For example, even though ϕ is definitely an irrational number, it does manifest a certain amount of regularity, such as certain digits recurring at precise intervals. As intriguing as those things are, I need to refer you to Livio for details. If I don’t observe some limits, this series would still be going a year from now, and—who knows—even the spammers might be scared away by then. So I’m going to continue to focus on ϕ as much as I can, and talk some more specifically about the relationship between the Fibonacci series and ϕ.

I’m going to introduce a new feature in this entry. The post would become meaningless if you or I just skipped all of the details, but I think I’ll keep more readers interested if I show them what they can skip without missing out on all of the content. So, I will introduce the problem for the day, but then give you the option of clicking on a link that will take you right to the conclusion without going through all of the steps to take you there. You will see it shortly below as a green rectangle.

There is a formula known as “Binet’s Equation,” named after Jacques Phillipe Binet (1786-1856), who was actually not the first person to discover it, but apparently created more interest in it than other famous mathematicians before him. Now, I’m going to trot out Binet’s equation, and you may just be horrified when you first see it. It looks like a huge monster, perhaps reminiscent of one such as the StayPuft Marshmallow Man ® from “Ghost Busters.” We’re going to describe Binet’s equation, and pare it down until it becomes one simple calculation. Then we’ll use it to find out what the 12th Fibonacci number is.

[Added on Monday: When I was working on this on Saturday night I had no guarantee that it would actually work. To be more specific, I assumed that the formula was right, but I did not know if I was interpreting and using it correctly. If it didn’t work out, I had no one to ask, at least immediately, what I had done wrong. Thus, my emotional reaction at the end was totally authentic. I couldn’t believe that I had brought of both stages: understanding the formula (a good first step to clarifying it for my readers) and applying it in order to get the right answer. ]

Skip to ConclusionHere is the promised link that allows you to skip over the material that looks intimidating.


Here goes. This is Binet’s Equation:

Binet's Equation

Intimidated? You should be—but only until we’ve taken it apart and seen the simplicity that’s behind the apparent complexity.

Seriously, I’m going to go extremely slowly in clarifying this equation and highlighting all steps. Maybe you will realize that working with such formulas one does have to be careful, but it does not have to be “*cough* laborious *cough*, if I may tease John of LRT one more time (all in good fun, I hope he understands if he should ever read it). One of the nicest compliments I ever received on my writing was with regard to my commentary on 1 and 2 Chronicles, when the editor at B&H told me: “Win, you have made Chronicles come alive.” I don’t know if it’s a comparable challenge, but I shall try to make Binet’s formula come alive as well.

Binet's Equation

“Fn” stands for which Fibonacci number in the chain you wish to calculate, i.e. the “nth” number. This is totally of your own choosing. For now, we’ll try to use the formula to figure out the 12th Fibonacci number, a number we can easily verify by just adding up the Fibonaccis. Maybe later, then, we’ll try our hands at calculating a much larger number, perhaps F95, but first we’ll have to see if we make it through this trial.

Binet's Equation

The expression “for n ϵ ℤ” reminds us that n has to be an integers. The Fibonacci numbers per se are not irrational, and are inhabitants of ℤ. The irrationality of phi and squareroot of 5 comes into play insofar our answer may need rounding up or down to keep the solution within the ℤ circle as well. From here on out, we're going to dispense with that little tail.

Binet's Equation

Wow! What are we going to do with that messy clutter highlighted in red? Actually, you may know already what we’re looking at here once I let it stand by itself. It’s the positive root of our good old equation for ϕ:



Binet's Equation

So, we can substitute ϕ in for that formula, and life is beginning to look a lot simpler already. We now know that, when the time comes for actual calculations, we can insert 1.61803 as the value of ϕ.

Binet's Equation

A similar thing is true for the next packet that I’ve highlighted in red.

Binet's Equation

This is the negative solution for phi’s equation, and you may or may not recall that it’s value is the same as the negative conjugate of ϕ, but terminology aside, you may remember that it amounts to -0.61803. Rather than writing “-1/ϕ”, which would mean that we're building up fractions again, I’ll use an expression that’s equivalent, but a little easier on the eyes, ϕ-1.

Binet's Equation

Let’s not forget that, when we eventually subtract that negative number, we can substitute a “plus” sign for the two negatives—if we ever have to do so. The last thing I can point out before we actually install some numbers is that both ϕ and its negative reciprocal will be raised to the power of n.

Binet's Equation

Now we’re looking at the formula that we want to instantiate.

Binet's Equation

I know; I know. You’re probably still not buying into my claim towards simplicity. Okay, I have one more trick up my sleeve to try to persuade you, and it's a good one. There’s a corner we may cut without doing ourselves any serious harm (just don’t run with the scissors, please). That whole –(-ϕn) business is going to get extremely tiny, yea infinitesimal, so quickly that it can be pretty much treated as negligible for our practical purpose. I’m allowed to say that because this formula does have a practical purpose, namely to find a specific number in the Fibonacci series. (Alright, different definitions of “practical” perhaps.) If we give ourselves permission to leave off the negative part, we can actually calculate the “nth” Fibonacci number of the series with the simple formula: One multiplication (ϕ raised to the nth power) followed by one division, dividing the product by the square root of five.

Binet's Equation

We set out to find the 12th Fibonacci number, and now we’re at the point where we can put some numbers into that simplified formula. If our result comes out really skewed we’ll have to work a little harder. Let’s substitute 12 for n:

Binet's Equation

If you’re using a sophisticated program, you can just plug in the symbols. I don’t expect that too many of us carry a value for ϕ12 in our heads, but Wolfram/Alpha has given me an answer of 321.987… . We can use a value of 2.236 for √5. So we can take that number and divide it by the number that Wolfram has given us. The crucial moment is drawing nigh:

Binet's Equation

We do the division and get …


Let me tell you how nervous I am right now. As I said above, I have not done this before with all of the appropriate details. Before starting to write down anything formal, I gave it a rough once-over trying just to see if it might work, but now I’m really wondering. I’m sure you’re in suspense, too, about whether I worked it out so as to get a plausible result. Of course, it’s possible that it didn’t. I am fallible and prone to small mistakes that generate drastic changes in the outcome. That reminds me of a time when …

Stop Babbling

Okay, okay, I’ll stop dawdling. It’s taking me a bit to get past this approach/avoidance dilemma. Let’s take it to Wolfram then and have it do the last division for us. The expected answer was 144. Here is the result of applying the truncated form of Binet’s equation:

144.0015 …

I can’t believe it. I really, honestly did not expect anything nearly as close to 144. I’m stunned and almost emotional. This was a long and winding road. Thank you to those of you who stayed with me through the entire length of this entry. I know it was demanding. But isn’t that result a whole can of high-octane awesome-sauce‼ Back in my childhood days in Gymnasium, the math teacher, Frau Dr. von Borke, had us memorize the multiplication table for up to 20. Thus I realize that 144 is not only the 12th Fibonacci number, but also 122. We can’t derive a rule from that fact, but we can see another piece of that beautiful mosaic of numbers. When God built in numbers onto the universe he created, he not only gave it regularity, but he infused into it a beauty that we ignore to our own loss.

It’s too bad that it has become almost fashionable these days to promote yourself as someone who doesn’t get along with numbers, wearing that self-deprivation almost like a badge of machismo. Look at what you’re missing!

I wish all of my readers a day filled with beauty on many different levels.

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Friday, June 17th 2016


Making Progress

  • STATE OF EXISTENCE: Waiting with a happy face.

Thanks to everyone who tried the Tripod sites. They are working now. I still can't get into my file manager, but for the moment there are lots of ways of getting around that, and I shall be patient. Come to think of it, being patient is about all I can do. I can wait with a happy face or with a frowning face, but wait I must. 

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Friday, June 17th 2016


Lycos.com and Tripod.com in a Coma. Requesting Some Assistance!

  • STATE OF EXISTENCE: Somewhat Frustrated in a Pickle.

For some unknown reason Tripod/Lycos has been inaccessible for the last several days. I house most of my on-line pictures with them, as well the majority of my subdirectorysites, including the presently growing series on PHI  and the all-encompassing www.wincorduan.com,  I also cannot get to their administrative sites, and my e-mails seem to vanish in the ether. From what I've been able to test, the problem lies with neither my computer nor any browser.  I have no idea what's going on with them, and I hope that the issue will be fixed pretty quickly. 

In the meantime, a lot of material is missing, such as quite a few pictures in this blog. I will try to transfer to Bravenet as much a I can as soon as I can; much of it is stored in my "local" files that I can re-upload, but some things are in their appropriate form only on Tripod. Obviously, I can't transfer those until I get access to them.  I guess that if I do get access to them again, I won't need to transfer them, but it may not be a bad idea to have important thinks saved in several places. 

So, please, be patient with me about the absence of certain illustrations and some resource sites. And, if you happen to think of it, would you please from time to time check either one of these addresses, both of which will send you to slightly different versions of my home page:

 www.wincorduan.com or win_corduan.tripod.com,

and let me know if you actually got through.

I will do the same of course, but someone else may be the first to find that we're hooked up with Tripod again. 

Mr. Lycos or Mr. Tripod (or whatever your names are), would you please 1) acknowledge my existence, 2) fix this mess, 3) show me what I can do--if anything--to help fix this mess, and 4) give me a prorated refund for the sizable amount of money I spend on you. 

Thanks, everyone!

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Wednesday, June 15th 2016



  • STATE OF EXISTENCE: bleary-eyed

Examples of the occurrence of the Fibonacci series in nature abound, and so do their descriptions. Therefore, apart from referring you to some websites and simply mentioning a few examples, I will not focus on that aspect. It’s been done often and it’s been done well. It’s the mathematical relationships that I’m trying to clarify.

“John” (no further name seems to be available) wrote an interesting article for the website, Let’s Reason Together, “It’s All in the Numbers.” He begins with the confession:

I’m not a mathematician. In fact, I generally have an aversion to numbers. They’re restrictive, require systematic, step-by-step (*cough* laborious *cough*) methods to manipulate, are terribly predictable, and generally unresponsive to creativity—at least, the right-brained sort (my sort) of creativity. 

I’m taking what “John” said there as a somewhat humorous expression of his humility. Part of what I’m trying to show here is that, even though working with numbers may require systematic step by step analysis, doing so properly also needs “right-brain” creativity. Furthermore, its laboriousness is not any more tedious than what I see many serious artists practice on a daily basis. Producing a work of beauty may necessitate long hours of intense labor. Take the Taj Mahal, for instance … Furthermore, if one wants to write on a phenomenon in math, one should ideally be on good terms with the subject, and I’m sure that most people would agree with that sentiment.

Taj Mahal“John’s” article mentions four areas of nature in which the Fibonacci numbers play an organizing role:​

1. the sections of pine cones and pineapples, which are arranged in a Fibonacci spiral;
2. the spiral arrangement of seeds within a sunflower;
3. the genealogical pattern of the male population within a beehive;
4. the relationship of the phalanges and metacarpal bones of the human hand.

We can add other examples, such as

5. the number of petals in a rose.

These are realities that Christians can point to and say that there must have been some intelligence involved in constituting the world in this marvelous way. Sure, we can find reasons why these arrangements are in the pattern of the Fibonacci series insofar as they contribute to the well-being of a plant, an animal, a geological formation, etc., but that observation only strengthens the argument that these examples are not random aesthetic or mechanical coincidences. If they serve a purpose, then there’s all the more reason to believe in an intelligent Creator behind the construction of the universe.

Sadly, once again I need to recite my litany about wishing that some Christian apologists were a little more careful when they assemble their evidence on behalf of God and the Bible. For example, the article by Fred Willson of the Institute for Creation Research, “Shapes, Numbers, Patterns, and the Divine Proportion in God’s Creation” [Impact: Vital Articles on Science/Creation,” #354 (December, 2002)], quite uncritically compiles what appears to be every claim for the golden ratio ever made, including some that are dubious at best, thereby undermining the credibility of what could have been a good case. E.g., his very first example consists of a correct description of the chambered nautilus, which he then labels, incorrectly, as displaying the “golden spiral.” (Note my earlier point that the mollusk in question does manifest a logarithmic spiral, but it does not fit the golden ratio as usually understood. At the time I just referred to this error in general; I had not yet seen Willson’s article. )  

I trust that John of CLR will indulge me as I return to indulge myself in the mathematical aspects of the Fibonacci numbers and the number phi. Theoretically, there need not have been a Fibonacci series in order for there to be a phi, since, as we have shown, the actual birth of phi occurred in geometry and number theory. The Fibonacci series can be described with an equation, in which Fn stands for the rank of a given Fibonacci number. What is the “nth” Fibonacci number?

(Fn = Fn-1 + Fn-2)

e.g., the 9th Fibonacci number = the 8th Fibonacci number plus the 7th Fibonacci number

So, we can just remember the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, … and fill in the slots.

34 = 21 + 13

This procedure is indeed lame, predictiable, and extremely slim on creativity. The matter of far greater interest is, of course, that the further we go, the closer the ratio of one number to its predecessor converges to phi. I realize that I have heretofore spoken of this relationship between Fibonacci and phi in a somewhat deprecatory manner, but, since you now realize that phi is not directly derived from the Fibonacci series, I can go back and endow the idea of convergence with the respect that it actually deserves. By “convergence” we refer to the properties of an ongoing series of numbbers whose numerical value comes increasingly closer to a particular number, ultimately coming so close to that number that the difference appears trivial.

John Derbyshire [Prime Obsession (Washington, DC: Joseph Henry Press, 2003)] presents us with a good illustration of convergence by contrasting it with its opposite result, called divergence. Most of this entry from this point on summarizes and maybe clarifies (if necessary) Derbyshire’s exposition. “Divergence” means that the value of the series keeps growing and, thus, eventually could be said to be infinity (∞).  He illustrates divergence by using the so-called harmonic series, which consists of the reciprocals of regular counting numbers (those that inhabit ℤ).

harmonic series

A close relative of the harmonic series does, in fact, converge. It’s as simple as re-creating the harmonic series, but using exponential powers of 2 as the denominators:

reciprocals of powers of 2 series

For a moment there you might think that, similar to the harmonic series, this one will also extend to infinity, but it doesn’t work out that way. To be sure, the number does keep growing; however, the rate of growth declines rapidly so that, as you go further along the best you can get is near-identity with a finite number. Let’s see what happens when we add it up this far:

fib ratio series

That’s pretty close to 2, and the further we go, the closer we will get, though we’ll never truly reach it. (Is anyone else reminded of the sizes of wrenches and sockets in an American-style set of tools?) We can say that the formula approaches 2, and that’s good enough for many purposes.

fib ratio series

Similarly, the ongoing ratio of the Fibonacci series approaches phi.

fib ratio series

and that, too, is good enough for many purposes.  

One such purpose is to help us determine the “nth” Fibonacci number when it’s a long way up the chain, say the 95th, without having to memorize the entire set of Fibonacci numbers or go through the tedious and (*cough*) laborious (*cough*) method of generating the lengthy chain of 95 links. But the clock, my energy, and your patience all lead me to realize that I better save that excitement for the next entry.

(Also, this is the first time that I’ve availed myself of MS-Word’s equation feature, which is pretty nifty. However, I wonder to what extent these formulas are going to translate into html. If I thought that thumb crossing had any value, I’d ask you to figure out how exactly one crosses one’s thumbs and then maintain your digits in that position.  -- Alas, it didn't work, so I needed to convert the equations to pictures for now.)

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Friday, June 10th 2016


Bring on the Rabbits!

  • IN THE BACKDROP: The hum of a new air conditioning system

Well, I've played my little game long enough, and I thank you for your patience. But now the time has come to start counting lagomorphs.

I hope that in my preceding discussion I have made it clear that the so-called golden ratio number φ (phi), has a life of its own, apart from the Fibonacci series. If you have caught on to that point, we are now ready to talk about the Fibonacci numbers and how they are intertwined with phi. (If you still say that phi is derived from the Fibonaccis, you're probably only teasing me, or you didn't read the preceding posts on the topic.)

Leonardo Bonacci (1170?-1250?) made a number of important contributions to math, the most significant of which is undoubtedly his promotion of the Indian/Arabic numerals in Europe, which made life a lot easier to anyone having to undertake any calculations. If you've heard of Leonardo at all, it was probably under his nickname, Fibonacci, and most likely in connection with the series of numbers that is designated after him.

The series is embedded in a puzzle that Fibonacci posed to his reading audience (Livio, 96):

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Please follow along with the picture as I walk us through the beginning of solving this puzzle. I might clarify, by the way, that we are counting the mature pairs only, just because that's how it's presented. The outcome would be no different if we counted immature pairs alongside the mature ones, except we would have only one numeral 1 at the outset, rather than two of them.

Fibonacci's Rabbits

  1. During the first month we have only the 1 original pair.
  2. In the second month the original pair will have produced one pair of offspring, which is not yet mature enough to produce any baby rabbits, so we are still holding at 1 mature pair.
  3. By the the time of the third month, the original pair will have given us another immature pair, and the earlier pair will now have matured, but not yet yielded any new rabbits. Thus the number we can put down for the third month for mature rabbits is 2.
  4. Come month 4, there is more offspring: one pair each from the original pair and the second generations pair. Also, the offspring from the newer pair will have matured. So the talley of mature rabbits for that month is 3.
  5. In the fifth month we get 5 mature rabbits, and each of the three that had matured in the previous month have contributed an as-yet immature pair.
  6. In the sixth month, the three previously immature ones will have matured and can get added to the previous five. Thus we have 8 mature pairs and five pairs that are only a month old--but probably can't wait until they are grown sufficiently to add to the pool.

I trust that you can see the pattern that is developing. Month by month the collection of rabbits is increasing. The numbers of mature pairs that we have now for the first six months are:

1 1 2 3 5 8

Each new number is created by taking the last two and adding them. So, let's continue this pattern until we reach the twelfth month, which will give us the answer to the specific puzzle that Fibonacci posed:

1 1 2 3 4 8 13 21 34 55

So, there will be 55 pairs of rabbits (110 individuals) living in the man's enclosure after a year. That doesn't sound too bad, but keep in mind that we are now entering a phase when the numbers will grow quite rapidly. Give it another three months, and the number will be 233 pairs, which is to say, of course, 466 grown rabbits. Then there will also be another 144 immature pairs (288 young rabbits), giving us a total of 754 rabbit hopping around, looking for lettuce and carrots.

As fans of the Fibonacci series know, and as I have intimated already, there is a close relationship between the Fibonacci numbers and φ. Take a number in the series and divided by the previous one, and you'll get some number that is in the vicinity of phi. So, let's look at two of such ratios.

1) The number of the twelfth month (55), divided by that of the eleventh (34): 55/34 = 1.617647059 ...

2) Since I brought up the fifteenth month, let's see what we get when we divide 233 by 144. The answer is 1.618055556 ...

Let us recall the approximate numerical value of phi: 1.618033988 ... . We can see that the longer we go on with the process, the closer we will get to phi. But there's something else that may not jump out at you immediately because of the many digits. Let us put these three numbers into numerical order, lower to higher:

1.617647059 -- 1.618033988 -- 1.618055556

The ratio of the 12th number to the 11th was below phi, whereas the estimate of the 15th to the 14th was a bit higher. I knew that would be the case, even before I tested it, just to make double sure. This is one of the interesting aspects of the relation of the Fibonacci series to φ: the different ratios of one number to its predecessor will always be an "approximation," though they get extremely close. But the direction of the error will alternate. The ratios with even numbers will be slightly lower than an more accurate rendering of φ, while the odd ones will exceed it just a little. Needless to say, as you have seen for yourself if you did the little exercise at the outset of this series, even with non-Fibonacci numbers the approximation gets very close, how much more within the Fibonacci series!

Next time: some more cool stuff about the Fib. numbers, as well as some theologial reflections.

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Thursday, June 9th 2016


Phi in the Sky

  • IN THE BACKDROP: Debra Whyte on StreetJelly.com

Yesterday was a gorgeous day in Hoosierland! Blue sky, sun shine, no need for an air conditioner at the moment. Today not so much. It’s supposed to get very hot again, so I’m glad we were able to make the arrangements to fund a new system. D. has already removed the old unit, will purchase the new machinery tomorrow on our behalf, and have it in place, hopefully, by the end of the week. --- Well, today that didn't happen. The funding is not yet showing up in our bank account. Hopefully it will be there tomorrow, so we can get the show on the road.


I’m going to continue with my discussion of phi. If this is the first time you happen upon my blog in a while, you may be a little confused as to the topic, particularly coming into the middle of an ongoing series. So, please know that the first four entries, which go all the way back to last year, are collated into a single document, to which I will add this one and the last two as soon as I can, so that it’s all in one place, and you don’t have to find your way backwards through my blog archive in order to read what came before.

Allow me to revisit what we’re doing here. We’re examining the nature of phi (1.61803…), the golden ratio constant, and for the moment we’re doing so as far away as we can get from the Fibonacci series and its exemplifications in art or nature. My point is to try to demonstrate its inherent elegance, which stems from its Creator just as much as the ­arrangement of flower petals and other “fibonaccied” items in nature.

Let’s go back to the question of two days ago: How contrived is the golden ratio? For example, take the golden triangle, which includes phi as an important ratio. Is it something that has been concocted just to show off phi, or is the golden triangle something that we can uncover and discover in other places. A partial response has been that it is included in the formation of a regular pentagon, and, thus, can be discovered without being guilty of fabricating an artificial instantiation just to smuggle in phi.

Phi is also found apart from geometry in number theory, which we can combine with a little algebra. Remember that phi is just one member of the infinity of all real numbers, which, as demonstrated by Georg Cantor, has turned out to be larger than some other infinities. However, it stands out from this uncountable crowd, right next to p, e, i, √2, and a few others, due to its special properties or notoriety. 

Here are two ways of finding phi without geometry. We will take a look at a couple of somewhat unusual-looking formulas and turn them into equations, which will resolve into phi. Here is the key: We came up with the formula for phi by setting the length of the base of a golden triangle as 1, and one of its sides as x. We were able to manipulate those numbers into a quadratic equation

x2 -x -1 = 0

The method for finding phi tonight is going to consist of finding several equations that can also be rearranged into the same quadratic equations. I found the following two equations in Livio.

Let’s start with a formula that consists of an endless fusion of square roots. Given:

 square root cluster

By the way, this thing is a formula, not an equation. You can only have an equation if there are two (or more things) that are considered to be equal, as, e.g., in

 square root cluster

It would be extremely tedious to work out a value for x from this equation if we want to continue to add the square roots of square roots, etc. But there is an easier way to do so.

We can square both sides. Squaring x gives us x2, and squaring the formula to the right resolves the first square root into a 1.

 square root cluster

Now it is apparent that the collection of square roots that follows after “1 +” is still identical to the original given formula since both extend to infinity,

 square root cluster

and, what’s more, we have already designated it as x above.

square root cluster

Then, substituting x for that nest of square roots, we get

which we can reformulate to fit the pattern we had looked for as

This is, of course, the formula which has phi as one of its solutions. Just think: we have derived it from that unwieldy collection of square roots.

So, now you feel like the Lone Ranger and want to save the world from more confusing formulas. That’s great. I’m with you all the way. Let’s try this monstrosity, which in mathematical argot is counted among a large group of “continuing fractions.” We start with a 1 and add the fraction, which has 1 for its first numerator and is followed by a never-ending, always-repeating denominator. Given:

Again, before we do anything else we need to turn this formula into an equation and label it with the variable x.

As you seek something wonderful in this equation, may I call your attention to the section underneath the topmost numerator of 1.

As in the previous equation, what you find there is actually identical to the originally given formula. There is no difference because both of them can be extended to infinity. And thus, we can safely apply the same letter variable, x to this new continuing fraction, which is also the old one. Remember now, that the area we have marked out is the denominator, and that the numerator of 1 still stands as before.

So, once we have substituted x for all of that clumsy denominator, we get a refreshingly simple equation.

Let’s get rid of the fraction 1/x by multiplying every term by x,

Then we have

and once more we can rearrange this equation into our favorite configuration:

Once again, we have turned a beast (the continuing fraction) into a beauty (phi). We have stumbled on yet another way of deriving phi without getting out our rulers or measuring tapes.


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Tuesday, June 7th 2016


Golden Rectangles


I’ll never be able to maintain a regular blog as well as take care of other things if I keep making myself include all kinds of diagrams and drawings. Still, it is fun, and for most of us a picture paints a thousand words—which is not to say that people who draw a picture necessarily forego an additional thousand words (ca. 1050 for this post). I started this entry yesterday (Monday), while still in Pokagon State Park. It was a good day, too. I had my first trail ride on horseback of the year, and June and I spent lots of time outside and swimming in the pool. Now we just got back to Smalltown, USA, and we need to see what we can do to get the new a/c system installed. Also, of course, I’m trying to finish this entry.

Pokagon Swan

Before returning to phi allow me to mention that my next StreetJelly date will be my regular set on Thursday evening at 9 pm EDT. I had to cancel last week due to circumstances that weren’t entirely beyond my control. After all, I was not forced to go to the dentist. Regardless, I see no impedance for this week. I’m planning on doing a show of some of my original songs. That’s at StreetJelly.com.

Please let me remind you that all of the earlier discussions on phi (except the last one—for now) are collected in a single site so that you can read the previous sections in a sequence that makes sense.

I left off last night by showing that phi manifests itself when you bisect two adjacent angles of a pentagon and, thereby, create a “golden triangle.” Then we were able to give birth to more and more golden triangles of diminishing size by bisecting one of their base angles each time. There is also a process that gives rise to new generations of golden rectangles, as we shall see below.

golden rectangle

By now I’m sure you have figured out why this rectangle should be golden: the ratio of the longer side AB to the shorter side AD is the same as the combined sides AB+AD to the larger side AB.

Now, we can lop of a square of length AD from the one of the ends rectangle, and we have a smaller rectangle left. I have placed the square on the left side of the rectangle. There is no rule governing that placement, nor can there be, since one can always flip the figure without doing it any damage. I’m placing my squares so that I can use the ongoing generation of golden rectangles to make a specific point in a short while.  

golden rectangle

The remaining rectangle (EBCF) now has the golden proportions. Let’s continue the process and remove another square designated by EBHG, and we have produced yet another golden rectangle, answering to the name of GHFC.

golden rectangle

Are we done now? Only if you want to be. We can remove another square and enjoy the sight of golden rectangle GIJF.

golden rectangle

And let’s do one more and call it GILK.

golden rectangle

And so forth … This is another unique treat that phi brings to us: We can go on and on bringing out golden rectangles by removing squares from one of its side.

Let us now reverse this process by starting out with the smallest golden rectangle and adding squares to it so as to create a newer, larger one, which will yield another golden rectangle by means of the same procedure.

golden rectangle

It is at this point that the placement of the square takes on significance. If I were to continue the enlargement procedure indefinitely according to my pattern, my arrangement will give us a spiral. In order to turn the tiniest of our rectangles into the next largest size, we’ll put a square underneath it. To reach the next size, we can place our square to its right. Moving on the next larger one, we can place the square on top of the one we have. Finally, to reach the largest size with which we began, we can expand it by means of a square on the left. Again, there is no point at which we have to stop, except for intrusions into our mathematical world, such as lack of available bandwidth, old age, or boredom. What you see is the beginning of a spiral. If we were to continue the process, the sequence can continue with the pattern of adding squares: down, right, up, left. Each time we get a new rectangle, it’s a golden one, and each one stands in proportion to all of the other by multiples of phi.

Now, let’s go back to the golden triangles and envision a similar spiral of them, increasing in size each time by the factor of phi. If we do so, we begin with a small golden triangle, and each time use one of the sides as the base of a new one. The animation below gives a brief sketch of the beginnings of such a spiral.


Are we still in math or, more specifically, geometry? Yes, we are. However, having shown you these constructions, I cannot forego mentioning one example of where we some people believe that they can see such a building process in action in nature, viz. an increase by a factor of phi in each particular stage.


The item in question is an animal belonging to the group called cephalopods [“feet on the head”], known among his friends and family (e.g. the octopus) as the chambered nautilus. This animal sets out its life in a rather small shell, but as it grows older and bigger, it manufactures larger chambers for its comfort and convenience—modern living for mollusks! A common belief is that this growth occurs according to the golden ratio. Unfortunately for ardent enthusiasts of the golden ratio, this is not so. It’s really a shame because once upon a time even your bloggist, who holds a B.S. in zoology, had been a victim of the same misconception. Gary B. Meisner, who calls himself the “phi-guy” and maintains a quite sizable website devoted to the golden ratio, explores a number of ways in which phi could be found as part of the proportions in the chambered nautilus, and leaves us with the ambivalent answer that, as yet, no nautilus that fits the pattern has been discovered, though it could not be ruled out that it will. For my purposes (and Meisner’s, if I read him correctly), that’s not enough to marvel at the golden ratio in nautili, though it leaves plenty of other things about which to marvel.

Chambered Nautilus

There are other, much better, examples in nature, but I’m once again postponing writing about those. There’s still too much I want to share with you about phi itself and where it shows up in its own world, the cosmos of numbers and formulas.


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Monday, June 6th 2016


Direct from the Pentagon


This entry comes to you from a state park in northern Indiana, where we are spending a few days. In order to make sure I get this installment out and start to establish a regular pattern again, I’ll save our adventures here so far for a later time.

Some Quick Catching Up

With regard to our health in general it’s been up and down for both of us, slightly down for June due to her fibromyalgia, slightly up for me because the expected neurological symptoms are not progressing as expected, and I’m extremely thankful for that fact.

If you’ve been seeing my various entries on Facebook, you know that I’ve been putting in a lot of time making music on StreetJelly.com as well as honing my drawing skills a little bit, the latter mostly for the sake of various websites. Long-time readers know that some of the websites I work on require use of pseudonyms to protect our readers (I’ll be happy to explain that idea if it doesn’t make sense to you), so I’m afraid that I can’t just provide a link from here under my name in order to show off my creativity, such as it is.  

I might mention that, during this blog hiatus I have made one new video, namely the Buddhist story of “The Death of Kisa Gotami’s Son.” 

I’m in the process of producing a Christian commentary video to go with that narrative. The graphics for it are already pretty much in place, and I just need to put together its narration.

After writing the entries on militant Islam, I was really beginning to feel burned out with the blog. I had no intention of dropping it then, and I still don’t, but I felt that I really needed a lengthy hiatus. Contributing to that sentiment was the fact that the various issues in the air, e.g., the presidential races and other social hot buttons have been matters on which I’d just as soon not get too caught up in the never-ending discussions, except for the occasional zinger on FB. (Please note: I didn’t say “mindless chatter,” because not all of it is, but much of the current discussions run in vicious circles that seem to do little more than to generate a lot of unwanted centrifugal force.)

On Thursday our air conditioning went out. Apparently, the motor fused, the wires burned out, and there’s nothing much to salvage on the outside unit, which is more than 20 years old. Since the temperatures have been around 90, this was not a good thing at all. D. of Riteway Heating and Cooling has informed us that replacing the outside unit (which is unavoidable) involves replacing the entire system, due to the prevailing rules of the EPA.  So, as mentioned, at this moment we are in a state park in Indiana, this time a couple of hours north of Smalltown, USA, waiting for the good folks of RHC to find a good replacement at a do-able price.


And now back to phi. Wow! That was a long time ago. Yes, indeed, but I had never intended to leave off where I did. I collected the previous entries on this topic, going all the way back to last November, in one single website, entitled PHI—Let’s Get It Right­­­­, and I highly recommend that you at least look very quickly at what I said earlier on the topic, so that this new addition will make more sense to you. I left off with this question: It’s all very well to stipulate a “golden triangle,” viz. one in which the proportion of its base to one of its sides is equal to the proportion of one of the sides to the sum of the base and the side.

golden triangle proportion

And it’s great that this proportion is precisely phi with all of the properties that phi has to offer. But is this geometric figure anything other than something concocted by an ancient mathematician for the entertainment of his guests on long November evenings? Can we find the Golden Triangle somewhere where it is right in place, playing a significant role in geometry? Let's go to the Pentagon to find out!

Let me issue an insincere apology to anyone who may have been misled by the term “Pentagon.” I was, of course, not referring to the building that houses the U.S. Department of Defense, and so I sup­­pose I should not have capitalized “pentagon.” Could it be that I was deliberately creating an ambiguity to get people to click on my blog?


Not this Pentagon!

As long as I’m pretending to issue apologies, let me hand out another one together with a rain check for the Fibonacci series, which figures so prominently in discussions on this topic. It has its own formula, Fn = Fn-1 + Fn-2, which is just a compact definition of the Fibonacci numbers and will tell us nothing new once I’ve explained the series. There also is a formula that we can use to determine directly the value of a specific Fibonacci number at any given point in the series, but to do so, we already need to know the value of phi and its negative reciprocal. As I continue to insist, as amazing as the Fibonacci series it, it yields phi only by convergence, not directly. Nothing wrong with convergence, but it’s not what I’m after at this time.

So, instead, let’s look at a pentagon in geometry. It’s easy to draw one with modern software programs; let’s remember, though, that in geometry anything that we draw will always be approximations. On the one hand, that means that you can’t just solve a geometric problem by measuring the lines. On the other hand, we need not attempt to go beyond normal human abilities in drawing our figures. And if­­­­ they turn out a little lop-sided, that’s okay, they’re only attempts to illustrate something that some aspects of reality cannot be reproduced by a sketch with paper & pencil sketch or a computer program.

So, let us think of a regular pentagon, viz. a two-dimensional figure with five equal sides.

pentagon/golden triangle

Now let us bisect two side-by-side angles. For convenience, we’ll go with the bottom two in the drawing, those that are located at points C and D. They converge at point A.

pentagon/golden triangle

Surprise! We have found the triangle ADC, and it turns out to be a golden triangle. (I’m skipping how that fact is derived from calculating the angles involved. See Livio, pp.  if you’d like more on that topic.)

pentagon/golden triangle

golden ratio

which comes out to phi on both sides of the “equals” sign. But that’s not all. Let’s take one of the triangle’s base angles, bisect it and run a straight line to the edge of the new triangle. We get a new triangle, DGC, and it, too, has the coveted “golden” proportions, as indicated by the number phi.

pentagon/golden triangle


No reason to stop there. Go ahead and bisect another base angle of the latest triangle, and welcome another golden triangle (GHC) into the family.

pentagon/golden triangle

We could go on and continue the process, but let’s go no further and celebrate our new discoveries. 1) We have found that a golden triangle is not just something created ad hoc for the sake of accommodating phi, but it is a direct property of a regular pentagon. 2) A golden triangle gives rise to further, but smaller, golden triangles each time one draws a straight line bisecting one of the base angles.

It’s not just the golden triangle that has the property of reproducing similar smaller versions of itself. Next time, we’ll look at the golden rectangle and take our first outing into the physical world by considering the chambered nautilus.


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Sunday, May 1st 2016


A Festschrift for Dr. Norman L. Geisler


Time to get back to the old blog to talk about some things too long for a Facebook post.

The catalyst this time is that yesterday (Saturday) I received in the mail a copy of the long-awaited Festschrift in honor of Dr. Norman L. Geisler.

I Am Put Here

What is a “Festschrift?” you ask. That’s a German term referring to a­­n anthology of essays and recollections in honor of a scholar who has made great contributions over a long career. More often than not, it’s tied to a particular point in time, such as the person’s 65th birthday, or whatever. It’s a good thing that this one wasn’t attached to any special event because it took a whole lot longer to see the light of print than was anticipated by any of the contributors, let alone by Terry Miethe, who edited this tome.

Actually, this is the second book that has been called a “Festschrift” for Norm. The first one, To Everyone an Answer: A Case for the Christian Worldview (IVP, 2004), was edited by Frank Beckwith, Bill Craig, and J. P. Moreland. It bore the secondary subtitle Essays in Honor of Norman L. Geisler, and it contains good articles by good people. There are four of us who have made contributions to both books, Ravi Zacharias, the Brothers Howe (Thomas and Richard), and your humble bloggist. There are a lot of names on the front cover, including mine, though not as Win Corduan. For anthologies my pen name is “and others,” and so I show up on a lot of book covers.

As I’m poking just a little bit of fun, please know that, my silliness aside, in terms of content, the 2004 book is truly excellent. Also, the occasion of its release actually got me a one-time invitation to take part in one of Bill Craig’s annual apologetics conferences. The title of my article in the 2004 volume was “Miracles,” not entirely the focus of my scholarship, but a topic that I do enjoy writing on from time to time.

Part of that enjoyment comes from the predictable knee jerk reactions one gets from internet atheists who get outraged by my conviction that they’re not qualified to determine if, when, or how a miracle may have occurred. After all, they’re atheists. They don’t think that God exists; in fact, many of them are not at all hesitant to tell us in so many words that people who believe in God are intellectual Neanderthals. But surely, if that’s their position, then they would be really bad internet atheists if they thought they could decide whether God had performed a miracle at a certain time and place. You can’t have miracles without a God who does them. So my duty is to shield internet-atheists from violating whatever oath or other formality they may have taken to ridicule Christians and other religious people. Maybe they pledge their allegiance to Richard Carrier or the well-known subject mentioned in the opening verse of Psalm 14 . I wouldn’t know. Nobody likes an inconsistent atheist, and if they try to declare what Christians should believe about miracles, we owe it to them to clarify that their opinions are irrelevant and hopefully thereby keep them from looking silly. If you don’t believe in God, you may perhaps believe in magic, but not miracles. (Note: not all atheists are “internet atheists.”)

Sorry, I'm still stacking as always. Pop!

Alas, as good as the 2004 anthology is, it did not contain many of the features that typically come with a genuine Festschrift, such as personal memories of the person honored. Terry Miethe decided that Norm deserved something that really fit the typical pattern and took the initiative to edit a book that conformed much more to the usual expectations for a Festschrift. Terry and I both go back to the golden years of Trinity Evangelical Divinity School when its faculty was a veritable Who’s Who in evangelical scholarship. We both got our MA’s the same year, with Norm as our thesis director. Terry went on to get his first [sic!] Ph.D. at St. Louis University only three years later and not too soon thereafter a second one at USC. He also bears the official title of “DPhil Oxon cand.,” which means he has done doctoral studies at Oxford University. You know, that old school in England.

I probably still have Terry’s e-mail somewhere, in which he invited various folks who have had close associations with Norm to contribute to his effort. I won’t check it now, and I can’t even remember right now what year that was, but his words for the project were something along the line of “I will get this done or die trying.” It turned out that he came close to the second option. He saw it through during a time when his wife, Beverly, was fighting cancer, and he also has come down with health issues. If I understand things correctly, he’s had to take early retirement on disability shortly after taking an unpaid semester-long leave to work on this project. Adding the virtual inability to find a publisher for what turned out to be a 441-page tome added a lot of stress to his life, which he did not need. Amazingly, with God’s help, he made it! And we are thankful to Wipf & Stock for publishing the book.

I think I understand now why Craig, Beckwith, and Moreland got their volume published with apparently little difficulty, namely because they did not include much of the usual Festschrift materials, but basically stuck with the collection of essays. At this juncture in time, publishers apparently are leery of investing in Festschriften because they see them as having ephemeral significance. Once the occasion is over, people have a copy of the book as quasi-souvenirs, and there may be no further call for it. In your bloggist’s opinion, the potential publishers are probably right for many cases, but in the case of Norm Geisler, may have been just a bit short-sighted. I’m pretty sure that this anthology will have a long life. Norm Geisler has left a large legacy, and—let’s face it—there are some really good essays in that book.  

Speaking of the essays, different readers will have varying opinions of them, and I expect that some of the reviews will reflect their personal opinions on “Stormin’ Norman” as much as, or perhaps even more than, the value of the contributions. Many of the essays are framed by the authors’ reminiscences of how Norm affected their lives. Terry did me the honor of taking my remembrance and moving it into the first section, entitled “Tributes to Norman L. Geisler,” under the heading of “Biographical Reflections” (xxxv-xxxvi). There I recount how … Oh, did you really think I was just going to give it away here?

Aquinas by Scrivelli

For my article I chose a somewhat improved version of a paper that I wrote for Norm as part of an independent study in the philosophy of St. Thomas Aquinas entitled “Some Features of Finite Being in St. Thomas Aquinas” (169-191). He had taught a course on Aquinas a couple of years earlier, and it would not be on the schedule again for a while, so I worked through the Angelic Doctor’s writings relying on various good resources and the occasional consultation with Norm. This would not be the last time that I would be grateful for all of the years of studying Latin (which I would eventually even teach), and it certainly came in handy there as well as for my thesis and—much later—my Ph.D. dissertation.

I might mention here parenthetically that for my first year of teaching at Taylor U, my department head and the dean had arranged that my teaching load would be just a little lower than normal (at the time 28 hrs. a year), but I gave up the load reduction for the spring semester because the philosophy majors asked me to, please, give them a course on Aquinas. Like I always say, Taylor students have been and probably still are simply the best! (Yet another stack! :) )

Anyway, I remember vividly writing that paper. It would have been at the end of the first quarter of my second year at Trinity and, thereby, when June and I had been married for just a few months. We lived in Kenosha, WI, because the cost of subsistence in Wisconsin was so much lower than in Illinois and it was worth the longish commute. At the time the MA program in philosophy required of 72 quarter hours of class work (usually two full years) plus a thesis, theoretically written during the second year while also taking a full load of classes. Many of us squeezed the two years into three.

In my typical fashion I waited to write the paper until the very last moment, specifically two days before it was due. First, I needed copies of some of Aquinas’ works that were only available at the nearby St. Mary’s Seminary in Mundelein, IL. (The most amazing library I have ever been in, right out of a medieval setting.) Since I wanted to take the books home, inter-library loan was the means to procure them for a time, but that would usually take several days. So, I devised a method that caused the raising of a few eye brows. “We’ve never done things this way before” was the response I got on both Trinity’s and St. Mary’s ends. I suspect they may not have allowed it after my machinations either, but the surprise factor probably helped me out. With an honest-looking guy (with beard and long hair, of course) standing there staring innocently and pleadingly into the librarian’s eyes, I got approval on both ends. I asked the inter-library loan person at Trinity if she would sign the request forms for a few volumes, e.g. De Potentia Dei, then personally took the forms to St. Mary’s and picked up the books in the name of inter-library loan. I was warned on both ends that if I screwed up (undoubtedly not the expression they used), my days of utilizing library services at St. Mary’s were over, but I was good and returned the books before they were due back. (Nowadays all those works are on-line.)

Then I went home, stacked the books on one side of the desk and started typing the final text right then and there; no note cards, outlines, or rough drafts. I do not recommend that method, but 1) there was no time for any preliminaries, 2) I had a clear outline in my head, and, thus, I knew exactly where I was going and which books contained the content that I needed, and 3) I was a good typist (and I guess I still am, though word processing has taken the adventure out of it), so I knew that slow and inaccurate typing would not be a serious impediment.

The bottom line: Norm liked the paper, and wrote some comments on the front that I am far too humble to disclose. With its publication, I hope that further readers will also enjoy it and learn from it. If not mine, then surely some of the other essays.

Other than Norm’s name, Terry’s is the only one on the front cover. And that’s how it should be. No “stars,” no special mentions, and, best of all, no “and others.” Thank you, Terry, for your work on this project and for pouring so much of yourself into it.

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Wednesday, December 9th 2015


Islam Material Gathered

  • STATE OF EXISTENCE: Tired of this topic.

This post is merely intended to let you know that all of the blog material on Islam from the last month or so has now been gathered at one site. You may find it more convenient to read the entries first to last, rather than having to backtrack blog-style. 

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Wednesday, December 9th 2015


Isis and it Expansion, part 6

  • STATE OF EXISTENCE: Happy to be done with this project (if, indeed I am)

This entry is the straightforward continuation of what I posted just a few minutes ago (part 5). It won’t make any sense without reading that entry first. (I don’t know that it will with it, but I know it won’t without it. :)  I will provide further formatting in a little while. 

The Early Situation in General

After a period away from large population Boko Haram established itself in Maiduguri, the capital city of the state of Borno. Its adherents probably came primarily from Muslims in northern Nigeria, who perceived that their condition in life, frequently abject poverty, was due to having to live in a non-Muslim (many would say “anti-Muslim”) society. However, as Cook astutely points out (2014, 5-6), if so, it would still be an unwarranted stretch to ascribe rampant poverty as the cause or raison d'être for Boko Haram as an organization. The means for recruiting members frequently do not dovetail with the overall goals of a group. I don't know to what extent Yusuf used this strategy intentionally, but it is certainly often easier to rally people together under the hatred of a common perceived enemy than under a common cause to attain a positive goal. 

Ever since 2002 Boko Haram carried out armed raids, frequently in order to obtain weapons as spoils in order to arm themselves and to bolster their often-meagre finances, but also to create fear among the population as they destroyed products of "Western Learning," such as schools, churches, medical facilities, medical storage sites, and, yes, quite a few mosques.

Inadequate Responses

Inside of Nigeria resistance was provided almost entirely by the Nigerian police, aka Security Forces, and occasionally vigilante groups (Cook, 2014, 14). Because Boko Haram frequently crossed the borders into Chad and Niger, either for raids or to seek concealment, the governments of those two countries used their armies to keep Boko Haram out of their land and hair. But they could not have an effect in stopping Boko Haram's advances in northern Nigeria. The intended goal of the establishment of northern Nigeria (at a minimum) as a separate Islamic state was slowly starting to take on reality. Only one crucial military unit, in fact, the most essential one, did not get involved, and that is--no, with all due respect, not the United States Marines--the army of Nigeria itself.

Cook points out some apparently inexplicable aspects in this set of events. One such was the fact that the Nigerian government did not act to protect schools.

What is most interesting for the outside observer is that there does not appear to have been any serious security measures in place in any of these locations. For Boko Haram’s attacks to be defeated, there needs to be a system for guarding and alarm for isolated schools. It is unclear, when Boko Haram has set educational institutions as its target, why the Nigerian government and military have not responded with setting up the appropriate security measures (Cook, 2014, 12).

However, the most glaring enigma is the more general one included in the above quotation, that the Nigerian government just did not get involved in earnest until finally in the spring of 2015, i.e. this year. Cook asserts in 2014 that “the track record of the Nigerian military to counter Boko Haram has been a miserable one” (2014, 14), and I’m afraid that he made a vast understatement when he wrote those lines.

The news agency al-Jazeera carried an article recounting the experience of a captain in the Nigerian army who had been sent out with about thirty men to rid a town of about 200 Boko Haram recruits. They received their orders and weapons (mortars/grenade launchers) with little notice and headed out to carry out their mission. When they were sufficiently entrenched and the combat commenced, the Nigerian soldiers to a man realized that their equipment was broken and the ammunition was out of date and spoiled. On the other hand, the Boko Haram fighters were relatively well-equipped and definitely trained in the use of modern weaponry. (I'll come back to that point below.) Somehow Captain X and his men prevailed by using their handguns and actually receiving reinforcements in response to their plea. However, that's as far as the Nigerian military was willing to take their mission, a few pro forma interventions here and there, but no significant effort, while the police and foreign military fought to contain Boko Haram as much as they were able. Cook describes the period from about 2010 to 2014 as a civil war in which only one side was fighting (2014, 29).

Some Significant Events

I am not going to detail the many inhumane acts carried out by the group over the last few years. Please see Cook's papers and other various articles on and off line. I must seriously caution you that if you're prone to physical reactions from descriptions of torture and other violations, such as the abduction of hundreds of 10-12 year old girls and their subsequent ordeals, you should not pursue the events in a lot of detail. Still, reality is what it is, and I do need to highlight a couple of occurrences over the years.

2009: The Execution of Muhammad Yusuf

The original leader of Boko Haram, as mentioned, was one Muhammad Yusuf. It became an open secret that, whereas he was expecting all of his followers to live a highly frugal existence, he enjoyed a life of luxury and indulgence (another enigma, but one that is widespread across many different cultures and religions). On July 25, 2009, the federal police in Maiduguri captured members of Boko Haram, which enabled them to track and arrest Yusuf. Cook (2011, 11-12) provides a translated excerpt of his interrogation in the police headquarters, which reads almost like a segment from Abbot and Costello. Yusuf was manipulative, mendacious, and verging on being insolent in his initial responses. Predictably, though not excusably, his captors reacted with increased anger. They humiliated and intimidated him, and eventually simply killed him without any further process of justice. A video of what happened is posted on YouTube, but I'm not going to link to it. There’s no need for us to get desensitized to brutality. If you really need to see it, e.g. for writing a term paper, it’s easy enough to find.

Needless to say, the execution of Yusuf caused a short pause in Boko Haram's activities, only to be renewed with increased vigor and a sizable increase in membership 2010 under the leadership of one Abubakar Shekau, a person who has made up for his apparent lack of knowledge both inside and outside of Islam by displaying unparalleled cruelty in word and deed.

Sidelight: It is interesting to me now that  in early November of 2009 I had a short conversation with a Nigerian Senator that touched on religion in his country. At the time I had no idea of what had happed a few months earlier, and my new acquaintance, clearly a serious Christian, obviously saw no need to bring it up. I attach no significance to that fact, except that I now realize how close I was to someone whose opinion I would have valued, had I only known about the situation and been tactless enough to ask.

Declaring a Caliphate

“Boko Haram” has always been a nickname, though its members and leaders used it themselves. However, in 2014, they changed their official designation to ISWA, Islamic State of West Africa, and stated that from that time on they constituted a caliphate. When I first read about that announcement, I tried to figure out whether it meant that ISWA was establishing its own caliphate under Shekau or aligning itself with the caliphate of Caliph Ibrahim of ISIS. Then I realized that much of the rest of the world also did not know what this new claim entailed either. I suspect that initially Shekau may have thought of himself as the ideal candidate, but since then ISWA has extended a request to ISIS to become a part of its team, and ISIS has accepted the offer.

Sidelight: One can find a number of good articles debunking the idea that the office of "caliph" truly has the great significance that many Muslims and Western analysts tend to ascribe to the term. From my standpoint as a scholar interested in precision, I appreciate such corrections; as a scholar interested in the role of religions in the contemporary world, once again I need to fall back on the fact that perceptions are frequently more important than theoretical accuracy in understanding the actions of people. (See, for example, Madawi al-Rasheed, Carool Kersten, & Marat Shterin, eds., Demystifying the Caliphate (Oxford: Oxford University Press, 2015).

A Bit of Fresh Air

In the short report on his studies of Boko Haram in the aforementioned alumni magazine, Professor David Cook, has finally felt free to sound a hopeful, if not optimistic note. After the elections on March 25 of this year, the government finally authorized the military to carry out a full-scale mission against Boko Haram, and has succeeded more so than many people had hoped for. To be more specific, as I understand it, the organization, which by that point was establishing territorial control over parts of northern Nigeria has been driven underground again. "Boko Haram's reign of terror is coming to a close at this time, thanks to a decisive response just in the nick of time," Cook rejoices and adds a little later that "it has been profoundly satisfying to hear that Boko Haram is finally on the run." Given the horrendous state prior to army intervention, one cannot help but to share his joy. Much oppression has been ended, and captives--particularly some of the abducted girls and women--have been returned, though bearing serious scars.

But Cook realizes that the hostilities are not over for good, and that Boko Haram still exists. Just as I thought about writing on this topic, I came across a report of a Boko Haram sponsored bombing in northern Cameroon. Furthermore, the Wikipedia article on the group contains a rather lengthy list of acts of terror ascribed to them this year, so after the momentary joy of some success, I'm afraid that we need to be more sober (it not somber) again.

Going Further

Writing in the context of exploring the options in American foreign policy in reaction to the events of Nigeria, Cook has suggested a rather cautious approach (2011, 23ff, & 2014, 24ff). Since clearly my opinion is far less informed than Prof. Cook’s, the fact that I concur may not carry a whole lot of weight. Our military resources are limited, and we do not have sufficient national interest in the fate of Nigeria as a political entity to get involved in one of their civil wars, even if we find one side quite repugnant. We cannot be the policeman of the world. A veritable crusade to West Africa to right all wrongs and establish a permanent Western-style government and economy would, without a doubt, wind up as disastrous as its counterparts in the Middle Ages. Nevertheless, there are some very good reasons to keep a close eye on what is going to happen in Nigeria in the near future.

1. Boko Haram, now ISWA, still exists, not just as an idea, but as a group of people who continue to pursue their goal, as made clear by the ongoing acts of terror.

2. Anti-government armies in Africa often are made up of a large number of mercenaries who move from country to country, offering their services to various causes. They provide experience in war in addition to bolstering the numbers of insurgent combatants who are capable with modern weaponry. The experience of Captain X, according to the article I mentioned above, is an apparent case in point. Thus, at least theoretically, the number of potential warriors on behalf of ISWA could actually exceed the number of Nigerian members of the group. Given sufficient inducement, ISWA could make a large-scale comeback, in which case the United States may have to get involved simply to continue the “war on terror.” Obviously, I have to leave it with “could” and “may.” Despite the statements of our country’s highest leader in foreign policy, we have yet to neutralize al-Qaeda, making it meaningless to say that we’re going to “do” ISIS in a shorter time frame. Any statements concerning ISWA suffer from even greater uncertainty.

3. We have begun to feel the impact of ISIS in the Western world, including even on our shores. I began this lengthy essay with the observation that ISIS is at the moment territorially restrained, but that it is also still expanding and increasing its influence (and President Obama recognizes both points, at least implicitly). Hopefully you may now be able to see one key to understand this conundrum. ISIS is expanding geographically because other jihadi organizations may (explicitly or implicitly) seek to make common cause with them.

I have tried to give an intelligible account of how this method has worked out in one West African country, and there's no good reason to expect it to end soon if it turns out that potentially new members of the intended global IS are receiving benefits from claiming affiliation with ISIS.

4. In the meantime, ISIS definitely gains from such expressions of support. Let me clarify.

The most effective strategy for guerilla warfare consists of spending much time in hiding, launching surprise attacks on the enemy so as to inflict maximum damage in a minimal amount of time, and returning to concealment quickly enough that the enemy has little chance to respond. To do so, one must have bases from which one can launch these sorties, and groups such as Boko Haram, even with few facilities, can provide just that kind of opportunity.

Is Boko Haram, given its recent losses, going to carry out terrorist attacks in Western Europe or the United States in the foreseeable future? Probably not. In contrast to ISIS, they appear to be short on resources and often look for financial support by means of plain old bank robbery.  

However, if they remain a viable, albeit underground, organization, they can still be of great assistance to ISIS, which has now displayed its capability for destruction on a world-wide level.

In short, in my debatable opinion Boko Haram/ISWA considered by itself should not be seen a threat to us here in the U.S. However, since they are probably once again at a point of desperation and presumably continuing to crave a role on the world stage, we need to take it seriously as a relatively new arm of the movement towards a world-wide Islamic State, as currently led by ISIS. Given the rhetoric of the last six or more years, they may very well be available to ISIS as a second-ranked group that is willing to do virtually anything in order to receive acceptance among other Islamic militant groups.

Do I mean, then, that the U.S. should send troops along with drones and bombs to Nigeria in order to root out what remains of Boko Haram? Of course not. I don't see where we could contribute anything other than creating a lot of bad will among all concerned parties—governments and insurgents alike—with only a minimal probability of success. If I can make a suggestion, it would only be the obvious one of keeping a close eye on northern Nigeria with a view towards exposing, if possible, significant contacts with other Neo-Kharijite and jihadi groups or a visible influx of potential mercenaries.

All of those prognostications aside, my overall purpose here has not been to advise what should be done, but merely to explain one means by which ISIS can increase its power as a global threat. I hope my efforts towards providing a little bit of further understanding have not been entirely in vain. 

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Wednesday, December 9th 2015


Isis and its Expansion, part 5

This is the first of two entries that I’m uploading pretty much simultaneously. Once both of these two posts are up, I will start gathering all of the recent material into one website.

Introduction: Gratitude from an Ancient “Owl”

Barring any unexpected events or questions that merit a lengthy public reply, this entry should be the last one directly dealing with Islam. Exploring Boko Haram, its background, and its potential implications, has been a learning experience for me. And it has definitely not led me to marvel at the inherent goodness of human nature.

I mentioned Professor David Cook of the Baker Institute at Rice University before. Actually it was reading his summary in the glossy pages of an alumni pamphlet from the religion department that alerted me to the subject and its implications. [David Cook, “Boko Haram” Religion Matters 2 (Fall 2015):2-3.] ---Obviously, this is not the first time that I have profited from the teaching from a professor at this fabulous institution.---The insights he provided sent me off to pursue this track, and I’m glad that he motivated me to do so. As it is, the best I can produce here is a quick snapshot, when it really should take an entire movie of the length of “Gone with the Wind” to get a decent understanding of Islam in sub-Saharan Africa.  I still love learning, and I picked up more than I could accommodate in these entries, but I hope that what I have done will be helpful to you to see this dimension of Islam outside of the Middle East.

One of the things that I appreciate particularly about Prof. Cook’s little summary in the alumni magazine is that he did not seek to hide his feelings about the subject: the frustrations, the disgust, the disillusionment, and, at the time that he wrote the piece, the exultation over the fact that things were apparently starting to head in the right direction. Sometimes the subjective reactions become a part of the facts without which an objective account is incomplete. (It was at Rice that I really learned to appreciate the value of phenomenology.) Prof. Cook’s accounts consist of two pdf articles under the aegis of the Baker Institute: Boko Haram: A Prognosis, 2011 and Boko Haram: A New State in West Africa, 2014. They are available as pdf articles, so it is easier, once one has downloaded them, to refer to them simply by their year and page number, rather than to insert hyperlinks each time. To a certain extent they are the backbone of my summary, supplemented by other books and articles. However, and I don’t mean this just as the traditional formality, whatever errors I have committed or am committing are truly my own fault, and, as always, I am happy to receive kindly worded constructive critiques.  

The Protean Identity of Boko Haram

In the last entry I brought up the idea, promoted by a number of scholars, that establishing shari'a (Islamic jurisprudence) in the northern states of Nigeria was unsuccessful partially because it was unenforceable on local levels as well as unenforced by the federal government. Of course, a group seeking to attain that goal could possibly defy the government and take enforcement into its own hands. The group that has become known as Boko Haram is a case in point. 

Aside: If you pronounce the name as “bokoe haraam,” you’re close enough to an acceptable way of saying it. Haram is indeed the same word as “harem,” that is used to describe traditional women’s quarters, and I don’t advocate changing how we say that word in English, but we should make an effort to get closer in this context. In a video to which I make reference below, I heard it pronounced in this way.

So, who is this group? What are their core beliefs? What scholarly traditions of Qur’anic interpretation do they follow? We cannot say as much as we would like to in answer to the last two questions, not because we don’t know, but because there seems to be little to know. It appears to me that, beyond a rigid commitment to Islam as an unclarified exclusivist ideology and the drive to eliminate everyone who does not accept them as the only true Muslims in Nigeria, Boko Haram has shown itself to be a group in search of a permanent identity. If we try to understand its rather thin ideology, we can only do so by, first of all, recognizing that perceptions usually outstrip reality in radical movements, religious and otherwise.

A general classification of this group places them amidst other groups that are called Salafi-jihadi (Cook, 2011, 2).  The second part of the label, "jihadi," is clear enough: they do not shy away from using armed violence to promote their cause. "Salafi," as I have mentioned in the past, is not as easily accommodated. The basic meaning of the term is based on a particular perception of the first four caliphs (Abu Bakr, Umar, Uthman, and Ali ibn Talid), who were known as the “Companions" (of Muhammad). They have been considered to be "rightly guided," and their teaching is treated as the only true and necessary understanding of Islam. But the content derived from their teaching and role modeling is not necessarily the same for groups that use it.

I noted in chapter 5 of the second edition of Neighboring Faiths that the Muslims of Sau'di Arabia reject being called Wahhabis and prefer to be called Salafis. Although I usually have little problem referring to groups of people by their preferred appellation, I said that in the case of the Wahhabis, the term Salafi is neither helpful nor really appropriate since there is no denying that the origin of their version of Islam lies with Muhammad ibn Abd-al-Wahhab (1703-1792)  and that, for that matter, they appear to follow the Hanbalite school of sharia. Here we see another reason: the term "Salafi" has been given to various other neo-Kharijite groups, including Boko Haram, who actually reckon Wahhabis with those who should be killed since they are tied to current power structures. The term Salafi simply does not say as much as might have been intended. In looking at whatever declarations and transcripts of interrogation I have been able to find, it strikes me that the leadership of Boko Haram has been steeped in folk Islam, but does not have a strong background in the study of the Qur'an, let alone its history of interpretation. I don't think that it is necessarily going too far to say that Boko Haram is driven by the desire to be seen as a major player in the Islamic world and might just be willing to accept any label that treats them as a major Islamic terrorist group to be reckoned with.

Even more than other Muslim groups, Boko Haram’s biggest target has been "Western learning," pejoratively called BokoHaram means forbidden, so this group became known under the negative label, "Western learning is forbidden.” Its original name was probably Jamā'atu Ahli is-Sunnah lid-Da'wati wal-Jihād, "People Committed to the Prophet's Teachings for Propagation and Jihad.” Cook (2011, 8) transliterates the name as Jama`at ahl al-sunna li-da `wa wal-l-jihad and clarifies that this choice of name (i.e. Jama`at) would place them alongside other similar groups in various parts of the world. The overall strategy of a jama group is to postpone overt violence until the movement has spread itself throughout an entire country over a long period of time. Since Boko Haram’s existence began almost immediately with violence, this formal name does not truly reveal the nature of the group. We need to consider another name that the group adopted a little further below.

In the absence of finding much material concerning the conceptual basis of Boko Haram in its early years, it seems to have become customary—if not obligatory—to present the following statement made by Muhammad Yusuf in a rare interview on the BBC:

There are prominent Islamic preachers who have seen and understood that the present Western-style education is mixed with issues that run contrary to our beliefs in Islam. ---Like rain. We believe it is a creation of God rather than an evaporation caused by the sun that condenses and becomes rain. --- Like saying the world is a sphere. If it runs contrary to the teachings of Allah, we reject it. We also reject the theory of Darwinism. (Cook, 2011, 8)

Hardly the stuff to go to war over, we might say. But things do run a little deeper than that. Abu Zayd, a frequent spokesman for Abubakar Shekau (Cook, 2014, 4), the group’s new leader after Yusuf’s death, explained that “Western learning” actually means "Western civilization," the perceived acme of everything that is false and perverted, in contrast to the purity of Islamic learning and Islamic civilization. Well, as Christians, we might be tempted to try to build a bridge and respond that we, too, do not like the immoral aspects that have made themselves at home in contemporary Western culture. But the term "boko" includes some items that we might want to acclaim as some of the better advances in the history of humanity, such as government by democracy. As far as I can tell, from Boko Haram's perspective democracy is not just a form of government, but also a deliberate means rationalize away the need for an Islamic  . And, furthermore, unfortunately a bridge is not of much use if it does not connect both shores of a river, and Boko Haram’s mission can be visualized as doing away with any potential bridge heads. Their policy, called takfiri, falls in line with other neo-Karijite groups, as it refers to cleansing Islam of impurity by use of the sword, but exceeds most of them in their brutality and lack of discernment. 

Concerning present rulers, Abu Zayd put forward the following statement, which astute students of logic could use as an example of the fallacy of composition and division.

This is a government that is not Islamic. Therefore, all of its employees, Muslims and non-Muslims, are Infidels (Cook, 2011, 11).

Since the United States is (rightly or wrongly) seen by many people as the strongest bulwark of Western democracy, it is the archenemy, even if it does not have a strong palpable presence in Nigeria. The Qutbi doctrine of the illegitimacy of any contemporary government seems to be highly visible. In fact, several times Abu Zayd stated as a goal of Boko Haram to render the country “ungovernable,” (Cook, 2011, 11 & 18) so that then the solution of a genuine Islamic government will be the obvious one. However, their recent support of a present caliphate does not fit in with the true agenda of Qutbism, and, with all of this ambiguity, I'm not sure that any presently available word (Salafi, neo-Kharijites, Qutbi, etc.) really captures their nature.

Aside: As my ever-faithful readers know, I think that the term "fundamentalist," particularly if it is used to lump together evangelical Christians, Hassidic Jews, and Islamic terrorists, combines far too many cultural and religious forms to be anything but an empty label. Someone somewhere decided to use the term beyond its original setting, journalists and scholars picked it up uncritically, and they now debate its true meaning as an overall category as though “being fundamentalist” were an objective attribute claimed by all the of the groups in question. I would suggest that a better alternative is to drop using it outside of its original Christian setting where it has a very concrete meaning derived from the book series called The Fundamentals, published from 1910-1915. 

All-in all, it's probably not very helpful to place Boko Haram into any neatly designed conceptual box. Pace Hegel, in this case the real and the rational are not identical; or, more specifically for this case, the perception of the real and the rational do not necessarily coincide. 

The final installment should be up in just a few minutes. 

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Friday, December 4th 2015


Isis and its Expansion, part 4

  • STATE OF EXISTENCE: Tuckered out, as usual :)

The Latest Attack

Yet another terrorist attack, this one on our soil in San Bernadino, CA, and apparently carried out by a lovely husband-and-wife team—their names are Syed Rizwan Farook and Tashfeen Malik, but I’ll just call them Mr. and Mrs. Killer—who first made sure that their baby was safe before embarking on their rampage. What a beautiful role model for progressive parenting in the twenty-first century! At this point it appears to be clear that they acted on behalf of ISIS. Their target was the state agency that had been the source of Mr. Killer’s bi-weekly paycheck; among its tasks was to provide help to families with special-needs children. And the time of their attack was, of all things, a “holiday celebration”—you know, as in peace on earth and good will to all.

Mr. Killer did not make it to the office party; I imagine that his fellow employees reasoned that, since he was a Muslim, he did not celebrate Christmas, Hanukah, or Kwanza. Little did they know that he would show up late, accompanied by his wife, and that together they would create a veritable apocalypse. One gets the impression from the reports that the Killers simply fired round after round into the assembled folks, not aiming at any particular person, just making sure that they would leave as many casualties as they could before departing

Prayer in San Bernadino
People praying on an adjacent golf course.
Picture hosted by CNN

It took the FBI a while to decide whether to classify this event as an act of terror, which may have struck you as a bit irrational with more than a dozen people dead. But there often are significant reasons not to make that decision on the spur of the moment. In this instance, since this was Mr. Killer’s place of work, he might just have had a grievance against his employer, which, he felt, could only be rectified by means of a mass shooting of his colleagues. In that case, he and Mrs. Killer would still be homicidal maniacs, but not terrorists. In this particular occurrence, since the couple was killed, the immediate consequences of the label are not as visible as if they had been taken into custody. Just keep in mind the fact that terrorists are considered enemy combatants who receive neither Miranda rights nor habeas corpus nor pro bono legal representation from some hot shot lawyer who wants to make a name for himself by defending the indefensible. Moreover, the follow-up is going to be very different now since it has been established that ISIS was behind the event, and that it wasn’t just the case that Mr. Killer was angry because he had a private grievance against his employers.

A Suggestion on Our Response

I would hope that it is impossible for anyone who hears or reads of this incident not to feel a great amount of anger. Let me underscore once again that Islamic acts of terrorism should be considered acts of war. In 2001, some people were upset with President Bush for using the language of war with regard to 9/11. However, that particular matter had already been settled by Osama bin Laden himself in 1994 (a long time before 9/11/2001) when he declared a fatwa that all true Muslims should kill as many Americans as possible in the context of an ongoing war. He described the US as the head of a large association of anti-Islamic nations, and he asserted that eliminating the head would cause the body to fall apart as a result. Keep that image in mind when you hear about people in the sands of the Sahara, who may never have had contact with any Americans, declaim how much they hate America in general and each American in particular.

Still, I pray that, as we respond to these world events, we will keep from becoming like our enemies. I confess that at times I have allowed myself to dehumanize them in my rhetoric, but doing so is not good because it can turn a travesty into a caricature. The people we are looking at are neither sub-humans, nor beasts, nor demons, nor “monsters thinly disguised as humans,” as I put it recently. If they were beasts, demons, monsters, or other sub-humans, it would presumably be in their nature to commit atrocities. But terrorists (and there’s no need to limit ourselves to Islamic ones) are human beings, and by their actions they show that they are evil human beings, surpassing others in their capacity to do evil even among our fallen race. If we say that they cannot help but go around killing other people because they are Muslims, we are not only stating something that’s as untrue as it is mindless, but we are also implicitly excusing their actions. After all, people shouldn’t be held responsible for something that they are programmed to do. It is the fact that they are human and that they violate their humanity in the most despicable manner that makes them such evil persons. The gravity and the moral decrepitude that we see exhibited here is so devastating because these are human beings, like you and me in many aspects, but with a putrid, rotten, decadent sense of right and wrong.

Now, having had this attack occur on US soil makes it all the more urgent that we get a grip on what has transpired in far-away Nigeria. ­­­Eventually the connection will become evident.

Uthman dan Fodio: A Historical Precedent

Uthman dan FodioIslam came to Nigeria more than 1,000 years ago and settled in various areas, particularly in the north where it makes contact with the Sahara desert. As is true with any religion in any region, over time the actual faithfulness of the governments and of the people to Islam varied greatly. There were times and places of great commitment, and there were also times and places where Islam was practiced quite superficially. As a result, from time to time there were reformers who attempted to purify Islam, and sometimes doing so would take the form of a jihad, a “holy war.”

A very prominent reformer and amazingly prolific writer was a man named Uthman dan Fodio (1754-1817; be aware of different transliterations of his given name) who lived in Northern Nigeria. After his message of calling for greater obedience to the Qur'an was rejected, he became the leader of a jihad that ended up in establishing a caliphate in the region of Sokoto. At that time it bordered on the Sultanate of Bornu, another strongly Muslim state. The caliphate of Sokoto lasted from about 1815 to 1904, when it was abolished by Great Britain in the process of turning Nigeria into one of its colonies. Actually, to be more accurate, they allowed the caliph to continue in a role as spiritual leader, but stripped him of political authority. By that time Uthman dan Fodio had become a hero for many Nigerian Muslims for bringing about this allegedly true Muslim state almost a hundred years earlier.

Sokoto Caliphate

Furthermore, in the eyes of many Muslims in Nigeria, dan Fodio’s jihad and the resulting Islamic state represent the ideal that should be repeated and re-attained in Nigeria. They state accurately that the British had clipped their wings, and they exhort each other to work towards the goal of turning all of Nigeria into a Muslim state as exemplified historically in Sokoto.

Religious Governance: It's Not Easy

Before continuing this brief narrative, I need to make a point concerning the difficulty of setting up a religious state. There's a perennial issue surrounding any attempt to establish a country under the umbrella of a particular religion. Exactly how will one implement such a plan? One can make rules, but can one bring about devotion and piety by force? The Old Testament gives us a good case study. God’s Law was the foundational document for the Hebrew people, and it included many dimensions, all of them obligatory: criminal law, civil law, laws on worship and sacrifices, laws on ritual purity, and so on. A person not only needed to be righteous and considerate, but also to meet the strict requirements connected with the worship of Yahweh, which specified when, where, and how to do so. And yet God’s own people strayed from him time and again. [Please allow me at this point to skip the opportunity for stacking and to forego a digression on the topic of the United States as a “Christian nation.”]

Islam is in a different position than some other religions because one of its underlying concepts is the mandate to establish an Islamic state, as I tried to show in my last entry. Consequently, it has a clearer set of requirements as to what would constitute an acceptable government. --- Please keep in mind that pure Qutbism, as exemplified perhaps by Osama bin Laden, aims for the immediate abolition of all governments, so what I'm writing here applies to al-Qaeda only with some with some qualifications. --- First of all, the government must be constituted by Muslims. Cooperative jizya-paying Jews and Christians are allowed to reside in the state and practice their religion insofar as it does not interfere with the supremacy of Islam in any significant way. So far so good, but where do we go from there? Presumably one could mandate prayers and pilgrimages, allow only halal (“permitted”) food to be eaten, and proceed to spell out a specific list of what else is halal and what is haram (“forbidden”). Or, one could simplify the process and decree in one fell sweep that the country, state, or region shall be governed by shari’a—Islamic jurisprudence.

Government by Shari'a

Please note that I said “jurisprudence,” i.e. the process by which legal cases are addressed and settled. It is not a set of crimes and their appropriate punishments per se. Some American Christians are haunted by the specter of shari’a laws being imposed on this country due to an increase of the Muslim population, not a likely scenario. Still, frequently such expressions of unease are accompanied by examples, such as adulterous women being stoned, or thieves having their hands cut off. But that’s not how shari’a works. By itself it is a method, not a book of laws. Those examples, as alienating as they are to us, are supposed to be outcomes of judicial deliberation, not overarching rules. In fact, some of the most frightful instances, e.g. the scenario in which an innocent girl was raped and subsequently executed by her brother, are not the product of shari’a at all, but miscarriages of justice that should never emerge when shari’a is practiced conscientiously. Shari’a is a rather complicated matter, and it is not surprising that there are four Sunni schools of law, which I have outlined as a part of my site on “Groups of Islam.” There’s no point in my repeating what I wrote there, but, please, read that particular page or get the information from some other source before you say anything else about shari’a. And, I might mention that fatwas issued without a preceding trial by such Muslim luminaries as the Ayatollah Khomeini of Iran or Usama bin Laden are as contrary to shari’a as a sentence being handed down without trial in our legal system would be. Shari’a can be a problem for non-Muslims, but evil people who will perpetrate injustice will not be constrained by any system, and they are the real problem. I think they would be just as evil under British-style common law, Roman/Napoleonic law, or African elder/ancestor-based traditional law as under shari'a. 

When the British left Nigeria around 1960, the newly founded federation faced decades of instability, and, sad to say, the right balance has still not been found. Democracies and dictatorships have come and gone, and corruption has been commonplace. We’ve already mentioned the strife between Muslims and Christians as well as the agitation between different groups of Islam, particularly between the conservative Sunnis and other groups. By the last decade of the twentieth century, finally a consensus seemed to have emerged among Nigerian Muslims, namely to unite in bringing about the one item that could theoretically engender a truly Islamic society: Muslim states should be governed on the basis of shari’a. I don’t need to point out that Christians and other non-Muslim residing in these states were quite unhappy with the idea. In some cases there may have been attempts at a dual-track systems whereby Christians were judged by a civil magistrate and Muslims by a Khadi, utilizing shari’a. David Cook published a map of those northern states in Nigeria that helps us visualize the apparent (but only apparent) solidarity of the Muslim population of northern Nigeria in agreeing to shari’a jurisprudence.  

David Cook Nigeria

It didn’t work. Among the reasons that Cook adduces for this failure, two stand out: 1) Nigerian states do not have their own police forces. All police units report directly to the federal government, and the federal government was not interested in enforcing shari’a, especially since some of its conclusions were at odds with federal laws. So, the absence of any power to uphold shari’a in those states that tried to apply it pretty much guaranteed its failure. 2) The Sunnis of northern Nigeria adhere to the Malaki school of shari’a, and this school, the second most conservative of the four, does, indeed, prescribe stoning for adulteresses. Several cases in which women were prosecuted under that category received international attention, and, in light of the pressure from outside of the Nigerian states concerned, the fatwas could not be implemented. Of course, once one aspect of the law showed itself weak enough to be overturned, there was no good reason to expect the others to be any stronger.

Cook summarized in 2011:

With the failure of essentially any possibility of enforcing the adultery laws, and the very obvious failure of  most northern Nigerian states to enforce bans on alcohol and other non-shari’a activities, 10 years after the initial implementation of shari'a it is clear that no ideal Muslim society has resulted (Cook, “Boko Haram Prognosis,” 2011, p. 8).

And he added the ominous words:

Most probably the frustration felt by the Muslims as a result of that fact has led to the rise of Boko Haram, first in Maidiguri (the capital of Borno) and then throughout the Northeast.

You may never have heard of Boko Haram before, but they, too, have made the United States one of their targets of destruction. And here’s the thing we’ll pick up on next time: They have some powerful friends. 

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Wednesday, December 2nd 2015


Isis and its Expansion, part 3


We are working towards a clarification of ISIS and its satellites, using Nigeria as an example. 

Please forgive me for beginning this entry with a statement that I have made numerous times here and elsewhere. It is of utmost importance in understanding the nature of Islam and, consequently, a difference between Christianity and Islam.

Christians: Resident Aliens in the World

Christians are exhorted to live in submission (which is not synonymous with blind obedience) to whoever constitutes their government, while Islam is never fully realized until Muslims live in an Islamic state.  The apostle Peter reminds his readers (1 Peter 2:11, 12a HCSB) that they are no longer truly at home in this world.

Dear friends, I urge you as strangers and temporary residents to abstain from fleshly desires that war against you.  Conduct yourselves honorably among the Gentiles ….

Maybe it's a paradox at first glance, but it is precisely because Christians are aliens in the world that Peter urges them to be exemplary citizens wherever they may find themselves. Christians should recognize the authority of a government even if it is ungodly. Who would fit that description better than the Roman emperors of the first century AD? Still, Peter states in 1 Peter 3:13-15 (HCSB).

Submit to every human authority because of the Lord, whether to the Emperor as the supreme authority or to governors as those sent out by him to punish those who do what is evil and to praise those who do what is good. For it is God’s will that you silence the ignorance of foolish people by doing good.

As mentioned above, the word “submit” does not entail unquestioning compliance with a state if it demands participation in an action that is clearly contrary to scripture. However, in that case we may have to accept suffering, as Peter clarifies throughout chapter 4 of his epistle. In a democracy we should voice our criticisms of the government when it does wrong and take whatever actions are available to promote the right. However, the idea of a Christian theocracy is not found in the New Testament, let alone promoted in its pages. To be sure, we are looking forward to a true theocracy when Christ will rule on earth during the millennium, but that event will occur in God’s own timing by his initiative and his actions.  We are not exhorted to establish the kingdom of Christ on earth before his return. – I am fully aware that many American Christians either do not understand this fact or do not want to accept it, promoting a veritable "Christian" political hegemony. They have apparently not yet come to terms with our basic identity as temporary resident aliens in a fallen world.

Islam: The Ummah as Political Entity

On the other side of this coin, genuine Islam demands an Islamic government, a condition that cannot be fulfilled by a “Western style” open democracy. Toying Falola provides several quotations from people with different outlooks on Nigerian religious policy in the front matter of his book, Violence in Nigeria. The first such paragraph comes from a Muslim publication, which—ignoring the strident tone of this quotation—does represent a basic Islamic dogma. I suppose that I could soften this statement a little by using another noun other than “dogma,” such as “perspective,” “opinion,” “preference,” etc., but doing so, even if it were to sound “nicer,” would be tantamount to concealing a crucial part of Islam. The statement comes from the Muslim Student Society publication, Radiance.

How long should we continue this trial and error and groping for a workable system? Given the present trend, what chances do we stand to survive as a nation? Can we even survive the present mess? Perhaps more important, has there been any nation in history which flourished under thoughts, ideas, institutions and political culture which are not only alien but hold in contempt the history, culture and conviction of a great majority of its people … For Muslims nothing is acceptable besides Islam (Muslim Student Society, Radiance: A Muslim Magazine for the Contemporary Mind, October 1982, 1.)

An easy response to this assertion is that the Muslim conquests and the imposition of Islam on non-Islamic areas constitutes a perfect example of such a cultural/political upheaval and the survival of its people (particularly if we ignore the hundreds of thousands of slaves captured and sold during those times). In Nigeria the expansion of Islam in its most successful times included the eradication of previous religions, cultures, and principles of law and government. Islam has been in Nigeria longer than Christianity and Western secular thought, but it can no more claim to be truly indigenous than any other religion except for those associated with the local tribal cultures, such as that of the Yoruba. In any event, it is Islamic dogma that Islam is not fully established in any location until an Islamic government is in charge. The reality of this dogma is supported both by the historical example of Muhammad and the early caliphs (the Salafi), for whom the spread of Islam as a religion and as a political entity were synonymous. It is also borne out by various verses in the Qur’an (Yusuf Ali translation).

Basis in the Qur'an

In 8:11a, Allah instructs Muhammad to accept those of his enemies who change their minds and repent:

But (even so), if they repent,
Establish regular prayers,
And practice regular charity,—
They are your brethren in Faith.

Further on, in 8:29 we see how the new social order under Islam will come about:

Fight those who believe not
In God nor the Last Day,
Nor hold that forbidden
Which hath been forbidden
By God and His Apostle,
Nor acknowledge the Religion
Of Truth, (even if they are)
Of the People of the Book,
Until they pay the Jizya
With willing submission,
And feel themselves subdued.

There is much that one could say by way of commentary on these verses. Al-jizya refers to the "unbeliever's tax" that Christians and Jews are required to pay for the privilege of living in an Islamic state (while being exempt from the zakat.) For "pagans" and lapsed Muslims, there is no such provision, and we have seen many examples recently that some militant Islamic groups ignore the Qur'an's exhortation to make this allowance. Suffice it for us at the moment to recognize that the establishment of a society regulated by five mandatory times of prayer (al-salat) and the regular collection of alms for the poor (al-zakat), alongside the oppression of the People of the Book and their submission, would not be possible if Islam would not have governmental powers. 

A potentially better case for Nigeria to become a Muslim state is also illustrated in the above quotation with the phrase "a great majority of its people." If the citizens of Nigeria were overwhelmingly Muslim, then, perhaps the establishment of an Islamic system of governance would become a legitimate demand. However, given the available plausible data there is no Muslim majority in Nigeria as a whole.

Falola Numbers describing the Nigerian population are notoriously unreliable. (See the appendix of Falola, Violence in Nigeria.) Different segments of tribal, geographical, or religious affiliations are likely to report the highest possible believable numbers—and sometimes overshooting the plausible. One motivation for reporting higher figures may be that they may result in greater allocations from the federal purse. After looking at various sources for numbers and their plausibility, I can say this much with a certain amount of assurance: The two largest religions in Nigeria are Islam and Christianity. It is likely that Islam has an edge over Christianity with a population of over 40 percent of the people, but not a mathematical majority. With a similar degree of probability Christianity can claim a number that is also over 40 percent, but a few digits behind Islam. Such conservative estimates are not beyond dispute. However, we can be fairly sure that the larger the claims are, the less plausible they are as well. 

Furthermore, even if there were a decisive majority of Muslims in Nigeria, it would be made up of factions that are seriously antagonistic to each other. The largest split places strict Sunni Muslims on one side and all other Muslim groups, such as Sufi brotherhoods, Shi’ites, Madhiyyas, and syncretists, on the other. These groups differ in various ways too numerous to enumerate here, but, most importantly, they would not agree on what an Islamic state would look like other than a vague generalization that it would be governed by shari’a, the Islamic approach to jurisprudence, rather than by British common law. Even stipulating a demographic skewed heavily towards Muslims in Nigeria, it would still be doubtful that a majority of them would favor a Sunni caliphate over other forms of governance. 

Perceptions of the ratio of Muslims to Christians are likely to be affected by the immediate surroundings in which a person lives. Depending on one’s state of residence, one could fall into the trap of thinking that what applies to the present locality is true for the rest of the nation. As mentioned above, different regions of the country and different segments of the population are frequently dominated by either Islam or Christianity/Western-style secularism. Two of the most thoroughly Islamic regions in Nigeria are the states of Sokoto and Borno. Both of them have a lengthy history as Islamic countries prior to colonization. Sokoto was known in the nineteenth century as the “Caliphate of Sokoto,” and Borno was usually referred to as the “Sultanate of Borno,” but also took on the title of “Caliphate of Borno” a few times. Historically, the two countries were frequently at war with each other, but under British supervision became provinces of the larger entity of Nigeria and are currently separate states within the federation.

I am heading towards a closer look at the Nigerian group Boko Haram, once again following my pattern of spending a huge amount of time on background (not to mention digressions and stacks without which you wouldn't recognize my blog). Suffice it to say for the moment that Boko Haram originated in Maidiguri, the capital of Borno, and that its inspiration probably derives from the history of Sokoto.

More very soon. I really think I will get to the intended subject with the next entry.

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Tuesday, December 1st 2015


ISIS and its Expansion, part 2


As usual, I did not intend to wait as long between entries as I did, particularly since I knew pretty much what I intended to write. However, once I started to pursue sources of information, I got caught up in the chain of interlocking events and their accounts and felt that I had to learn a whole lot more before putting my thoughts into words. This post is only the beginning of what I’ve written over the last few days, and hopefully I will get another entry ready very quickly.

Bibliographical Note

In writing a blog entry, judging when and where to include references is not as straightforward as it would be in writing an academic paper, so I’m going to give you some of my best sources on the topic here and then provide links or abbreviated notes in cases of direct reference.

Toyin Falola. Violence in Nigeria: The Crisis of Religious Politics and Secular Ideologies: Rochester, NY: Rochester University Press, 1998); Nehemiah Levtzion and Randall Pouwells, eds. The History of Islam in Africa (Athens, OH: Ohio University Press, 2000). Both of these books were written before the public emergence of the Nigerian group known as Boko Haram, which is a positive point for me. It is all-too-easy to view history in the light of present events and, thereby ascribe significance to them that they might not have had at their time.

For the basic facts and some clear-headed analysis concerning Boko Haram I recommend a number of articles by David Cook, associate professor of Islam at Rice University who is working in association with the James A Baker III Institute for Public Policy. E.g. “Boko Haram: A Prognosis,” 2011, and “Boko Haram: A New Islamic State in Nigeria, 2014

In the last entry I stated that, to the best of my knowledge, President Obama was correct when he claimed that ISIS, the radical Islamic group, has been contained in its area of origin, namely Syria and Iraq. Still, Isis is growing in power and geographical proliferation. How is that possible? The answer is rather simple. In various parts of the world other radical Muslim movements are pledging their allegiance to ISIS and to Caliph Ibrahim, their leader. Recent events in Africa can serve as an example.

Maps hosted by InfoPlease

The African country of Mali was recently in the news because of the al-Qaeda sponsored bombing of a hotel. How many of us needed to look at a map, however briefly, to remind ourselves of the location of Mali? I suspect that there may have been quite a few people in this country—though obviously excluding my always-well-informed readers—who were not even aware that a country named Mali existed, let alone where on the globe its citizens pursue the quest for their daily handful of millet. How many among us knew or remembered that its capital city is called Bamako? Not I. I must confess that the only name of a city in Mali that I could have readily identified is Tombouctou (better known as “Timbuktu”), which once upon a time was a rich and thriving center of trade, one of several hubs on the route connecting the Arabian Peninsula with West Africa. Timbuktu also figures largely in the accounts of possibly the most outstanding 19th-century explorer of North Africa, Heinrich Barth. (For my German readers: Es ist unmöglich, die Berichte des Heinrich Barths zu lesen, ohne den Eindruck zu gewinnen, dass er dem Kara Ben Nemsi sozusagen Pate gestanden hat.) It is telling that in American slang “Timbuktu” has taken on the meaning of “way out there, about as far from our locality and civilization as you can get,” a characterization that unfortunately covers up its historical importance.

Maps hosted by InfoPlease

Well, we’ve become (re-)educated as to the existence and locations of Mali and Bamako. I’m not terribly grateful, though. There are better and pedagogically more preferable ways of learning geography than by tracing the activities of terrorist groups, but we have been forced into it. And, furthermore, we probably should work on remembering the names of African countries and cities. Events on that continent will very likely have an increasingly powerful effect on the rest of the world in the not-so-distant future. Let us move a little further south from Mali and look at what is happening in Nigeria.

 Looking at the figures given in the Wikipedia, Nigeria has become a country to be reckoned with when it comes to its impact on the world. Alas, unless it will find a way to establish political stability, a backward slide is a strong possibility. Centuries ago, Nigeria was the center of multiple kingdoms and several larger empires prior to European colonization. According to the numbers supplied by the Wikipedia, its total population now is 182 million which gives it first place in population size among African co­­untries and eighth place in the entire world. Similarly, its economy is (or perhaps has been until recently) ranked as the twentieth in the world and the largest in Africa. Given those numbers, what happens in Nigeria cannot help but have an impact onto the world.

Islam came to Nigeria a long time before Christianity, and it slowly worked towards the elimination of traditional religions. Its spread was due largely to various jihads, some of them in a sincere effort to purify Islam in Muslim territories and some of them clearly only as the pretext of some ruler’s ambitions to expand his land. Prior to direct European influence, the two basic options were either Islam or traditional religions, such as that of the Yoruba people, which was eventually exported to Cuba via the slave trade under the name of "Santeria" (see chapter 7 of my Neighboring Faiths.)

Not-so-aside: I’m really sad that there is a need to inject the following observation, and I’m not doing so in order to impugn Islam at this point, but simply to rectify a claim one frequently hears from Muslim apologists. They may tell you that Islam came to this country originally by way of Islamic people who had been captured as slaves and transported to America. Now, I cannot state that there were no Muslims among the slaves who were brought here. However, the reality is that the overwhelming majority of eventual slaves were non-Muslims who had practiced traditional religions, and who became the spoils of war during the various jihads. Furthermore, the chances are good that, insofar as there were Muslims who became slaves, they had a different affiliation from those who were victorious in their military campaigns. But this point is quite certain: Regardless of the make-up of the slave population, those who captured slaves and sold them to Europeans and American were almost entirely adherents of Islam. (And, of course, those who bought them in Great Britain and North America frequently called themselves "Christians," but we're not rewriting history to cover up that bitter truth. Also, it was Christians of various stripes, including among others Quakers and the larger-than-life John Brown, who were in the forefront of abolition.)

Christianity came to Nigeria for the most part by way of Europeans. Although the officially announced policy of Great Britain was to abstain from interference in local governance and religious practices, Nigeria was opened to Christian missionaries, who came and evangelized, aided by works of mercy, such as medical care and education. It was inevitable that many Nigerian people picked up Christianity, the religion of their more advanced colonial supervisors, in order to attain success similar to theirs.

The geographical and political divisions of Nigeria are complex and are increasing as ever-new states are created by a process of balkanization. But a general pattern emerged by the beginning of the nineteenth century that is still visible on a simplified map.

The three most important cities for our discussion are Sokoto (northwest), Maiduguri (northeast),
and Abuja, the capital (center).
Maps hosted by InfoPlease

Speaking then in overly general terms, the northern region of Nigeria is predominantly Muslim; much of the population consists of Hausas, a Berber group, and the practice of Islam has been quite orthodox on the whole. The smaller southern area is best seen as divided into three subareas. The Western area is home to the Yoruba, most of whom have now officially abandoned their traditional religion in favor of Islam. However, in contrast to the northern Muslims, the Yoruba have tolerated a certain amount of syncretism and greater diversity in Muslim practice. The Igbo, who are now primarily Christians, live in the Southeast; they dominated world news in the late 1960s because of their unsuccessful attempt to secede from Nigeria under the name of Biafra. The ensuing war and sanctions contributed to the death of an estimated one million Biafrans. Their territory is crucial to the entire country’s economy due to the presence of large oil reserves. Finally, between the two flanks there is the corridor labeled as “Midwest.” As I understand it, in this region one may find the largest number of well-educated people, who wish to model Nigeria on European states. Along with those aspirations comes at least a nominal commitment to Christianity.

No area is entirely “religiously pure,” and Christians and Muslims are found to various extents in areas where the other group dominates. Following patterns set in all too many locations around the worldminorities in a given region may be subject to discrimination and persecution. (Furthermore, as I had to learn in order to get a "feel" for Africa a couple of decades ago, we must always remember that traditional religion continues to live in the hearts of many people, even when it is not outwardly visible, but that's another story). After consulting a number of sources, including some written by Muslims, and allowing for a reasonable number of exceptions, I find that it is fair to say that Muslims in general have advocated turning Nigeria into a Muslim state, governed by shari'a, while Christians have maintained that Nigeria should be a secular state in which there is room for both religions. Due to the legacy left by the British, particularly immediately after independence, governmental positions have been filled more often than not by (at least nominal) Christians. 

More as soon as I can get it together. 

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Saturday, November 21st 2015


Isis and its Expansion, part 1

  • STATE OF EXISTENCE: Moving on fumes, but moving ...

 (Please note that “ISIS,” ”ISIL”, “Daesh” and “the Islamic State” are being used synonymously for the same entity, and I will give myself flexibility as needed.)

This series of entries is intended to provide some further information concerning the militant groups of Islam who are creating havoc around the world. In the first installment I’m going to refer to some remarks made by our president and secretary of state, but only to illustrate certain attitudes. This is not intended to be a political topic per se, though it has to start that way. My basic point should be non-partisan as far as American politics are concerned.

1. Correcting a Misperception.

I just ran across the site “Truth-O-Meter,” and it appears to do a pretty good job analyzing politicians’ utterances. I’m not sufficiently familiar with it yet to know whether it has a distorting bias, but alone their reproduction of direct quotes in context makes it helpful. Here’s an example:

When President Obama said on the very day before the Paris bombings that “ISIS was contained,” given the specific aspect he was addressing, he was correct. Maybe he was still riding high on the waves of the execution of Jihadi John and various other minor successes, but those things had nothing to do with that statement. If you look at the television screen of “Good Morning America” pictured in one of the T-O-M  pages, you will see a good illustration of how his statement was misunderstood,

President Obama Doesn’t Think ‘ISIS is Gaining Strength’

the little inset informs us. Taken in any way other than the specific context in which he made the statement in question, it would have been outrageously ignorant or deceptive. However, what the president was referring to was clearly limited to the geographical expansion of the territory ISIS is claiming within the countries of Syria and Iraq. For a while now, ISIS has not been able to stretch those boundaries, and that was what Mr. Obama was alluding to. He was quite aware of and acknowledged the growth of ISIS in territory and power in other parts of the world.

And that’s what I want to write about here, though I’m afraid I need to address some more matters beforehand.

2. Exposing a Misrepresentation.

On May 5, 2011, I closed my obituary of Osama bin Laden with the following statements (edited just a little), hazarding some guesses of what the future might bring:

Someone else will come along, maybe not immediately, to fill Osama's shoes, and the fact that we don't know yet whether they will be sandals, wingtips, or boots introduces a dangerous element of uncertainty. His death is a setback for al-Qaeda.  But only a setback.  There are too many people, too many cells, too many individuals whose brains have become pickled with Qutbite rhetoric for anything to have changed in the long run. I'm afraid that we have not heard the last of al-Qaeda and its nefarious activities. This is my opinion, and I can't imagine that too many people would seriously think otherwise.

Well, of course I was wrong. Some people have attempted to convince us otherwise. The Obama administration has attempted to use the assassination of bin Laden as a license to claim that they had overcome al-Qaeda.

Somewhat aside: One speculation concerning the pretzel of misinformation that we have come to know as Benghazi Gate is that President Obama allowed (or authorized?) the misrepresentation of the facts in that case because the reality, viz. that the miscreant behind bombing the American consulate belonged to al-Qaeda, would have distracted from this supposed victory. (BTW, Ms. Susan Rice, the former ambassador to the U.N., who spread the false cover story and then blamed the intelligence services, is now one of Mr. Obama’s chief foreign policy advisers.)  

One might have thought that, as things were developing over the subsequent few years, Obama’s people would have silenced their trumpets on this matter, but until this week that has not happened. In what I wrote at the outset of this post I agreed that the president’s statement concerning the “containment” of ISIS had been misconstrued, and that within its proper context it was correct and appropriate. No such exoneration is possible for the current Secretary of State, John Kerry, who said on the day before the bombings in Mali that “they” (the Obama administration?) had neutralized al-Qaeda. Here is the direct quote,

It took us quite a few year before we were able to eliminate Osama Bin Laden and the top leadership of Al-Qaeda and neutralize them as an effective force. We hope to do Daesh much faster than that. We think we have the ability to do that.  

The statement was preceded by a caution that dealing with these terrorist organizations takes time, and that people need to be patient. However, Secretary Kerry seems to know (I do not see how) that with regard to ISIS (Daesh) the time table will be faster. I cannot help but see his declaration as false, irresponsible, and illiterate.

Illiterate: What in the world does it mean to “do” Daesh? What kind of a Secretary of State uses such street jargon in a public statement?
False: The fact that al-Qaeda has not been neutralized as an effective force has been obvious to most of us for quite some time now and didn’t need to be confirmed by the bombings in Mali on the day after Mr. Kerry’s speech. The atrocities didn’t tell us anything new about al-Qaeda, but they did reveal that our Secretary of State appears to be out of touch with reality.
Irresponsible: Consequently, his promise that ISIS will be eliminated (if that is what “do” means) more expeditiously than al-Qaeda is not only groundless, it’s meaningless.  But it attempts to convey a false sense of reassurance. (One is reminded of his presidential campaign, in which he insisted time and again that he was a better candidate than George W. Bush because he had a "plan," but he never told us what the "plan" was.)

And, thus we come to the important question of how ISIS is continuing to grow and become a veritable rival to al-Qaeda.

I am beginning to realize that, once again, this matter is going to take more than one installment, and I am running out of steam fast. Working through the political material took more out of me than I had anticipated; it takes a lot of energy for me to write in a civil manner about Mr. John Kerry, the person and the politician.

Next time, I’m going to bring up certain events in Nigeria as a case in point of how ISIS is increasing in power and influence.

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Wednesday, November 18th 2015


The Golden Triangle and its Offspring

  • STATE OF EXISTENCE: Once again, pretty exhausted

We haven't had too many typical November days so far this year. There's been a lot of sunshine and it's stayed relatively warm so far--high 50s and low 60s for the most part. Today (Wednesday), when you look outside, it's definitely November. It's still surprisingly warm, but all the skies are gray, the leaves are gone, and there are four or more strong winds outside.

June has been struggling quite a bit after her last dentist appointment. Without going into details, it appears to me that they crammed what should have been two appointments into one, and she is still hurting pretty badly.

I have entitled my StreetJelly set for tomorrow (Thursday) night (9pm EST) "Blame it on the Bossa Nova" in recognition of some wonderful people in my audience who either live in Brazil or have had some connection to that country. Right now, I am totally resisting the urge to write about the distinctiveness of the Bossa Nova rhythm, but I'm intrigued by it, and so it will come up eventually. I'll undoubtedly mention it tomorrow night. (In the same context, I'm also refraining from writing about the Portuegese language, which has me fascinated from a linguistic point of view.) In addition, on that next StreetJelly set I will include some recognition of the people of Paris as they are grieving and hurting.

I do need to come back to the topic of ISIS very shortly and give you some additional information that you may not be aware of, though it's rather important for the entire picture. So, I may be oscillating between topics for a short while. Today I will slowly work up to the actual focus of these entries, the remarkable number phi (φ)


Perhaps you feel that I’m spending too much time and bandwidth on more or less relevant peripheral matters. But, you see, I’m trying to provide a background, as limited as it will be at that, which allows us to do more than simply gush at the remarkable presence of phi in nature. I guess one could view my effort here as trying to counter the attitude of "I don't know anything about math, but I know that I like phi." I teased you at the outset of this series with a clone of the famous Fibonacci series, and I will return to it. Actually, I haven't mentioned it yet directly, but knowledgeable readers recognized it immediately behind the thin disguise I gave it. However, right now I’m trying to disconnect phi from the Fibonacci numbers as much as is possible. The Fibonacci series is a remarkable function, and it is true that it converges to phi the farther you compute it. However, phi is not directly derived from the Fibonacci function—never has been and never will be. But, as I said, don’t worry; we’ll come back to it.

To the best of my knowledge, the first written record concerning the ratio that we now express with the number phi stems from Euclid of Alexandria (ca. 300 BC) in his best-seller The Elements, a book that stood for several millennia as the final authority in geometry. I will attempt to describe (please note—not prove!) Euclid’s discovery of phi, though taking a slightly different sequence of steps. (I’m following pretty closely on the heels of Mario Livio, The Golden Ratio, pp. 78-82.) But, as alluded to above, please make sure you realize that Euclid was not writing about a number, but about the proportions of lines in certain geometric figures expressed in words.

1. The Golden Triangle.

Let us examine some features of a triangle, which is sometimes called the “Golden Triangle.” Eventually I will tell you how we can derive such a triangle from another geometric form, but for the moment we can just assume that there exists such a figure among the many ideal geometric objects. It is a somewhat pointy triangle (“acute”) and its two longer sides are of equal length (“isosceles"). We’ll label the corners A, B, and C and call the three sides a, b, and c respectively. Let's also label the three angles alpha, beta, and gamma.

Gilded TriangleMy readers undoubtedly know that one feature of any (Euclidean) triangle is that its three angles add up to 180 degrees. The Golden Triangle’s base angles measure 72 degrees each, which leaves 36 degrees for the top angle. You see a picture of it on the right (copied from Wolfram Alpha and modified--or wantonly bedazzled--by me).

We could print out this picture, measure the sides of the triangle and, perhaps, come to some interesting conclusions. However, proofs in math or geometry based on physical measurements don’t count for much. Standards of measurement are invented by people, and you’re going to get different numbers if you use, say, centimeters rather than inches. Ideally, the relationships, such as the ratio of one measured line to another will be the same regardless of the calibration of our physical rulers. For example, 1 inch divided by 2 inches and 2½ cm divided by 5 centimeters both come out to ½, but the measurements on which this ratio is based will still depend on the accuracy of our instruments, which is always limited, and so we couldn't really be certain that the ratio we obtained is correct. By contrast, if ever in the course of carrying out some little household carpentry project, I wound up within 1/100 of an inch precision for all of my cuts, it would be a miracle. But in math the difference between 2.49, 2.50, and 2.51 can be crucial. There is no "level of tolerance."

However, we can resort to a mental ruler, whose fundamental unit is completely exact for our purposes, and on which we can all agree, regardless of what the usual units of length in our daily lives may be. We can posit that 1 mental unit equals exactly the length of b, the base of this triangle without taking account of its size when we drew it or even whether we drew it correctly. Obviously then, we should not be surprised by the fact that the length of b is 1 since we made it that way. Therefore, by assumption,

b = 1.

Having done that much, we still do not know the lengths of the two other sides, which are, of course, equal and, for the moment at least, we can only represent them with a variable, calling on the ever-prepared “x” to do its usual duty. [See note 1 at the bottom.] Thus,

a = x     and     c = x.

Thus, so far this is what we know about our triangle.


Golden Triangle

Length of b = 1
Length of a = x
Length of c = x

Angle α = 72°
Angle β = 72°
Angle γ = 36°

2. The First Proportion: a to b

Let us now figure out the proportion of one of the sides (we’ll pick a) to the base, b. Given the convenience of our mental ruler, the proportion ab  is equal to x1.

ab  = x1

Normally we would remove the useless 1 from this expression, since there is no difference between x1and x, but let’s allow it to stand in the redundant form, x1 for the moment because it will illustrate the ensuing point a little more clearly.

3. Opening Up the Triangle to Create Line g. 

Now we will open up our triangle.

We shall disconnect line a at point C and, using point A as a hinge, swing it downward until it has become a continuation of the horizontal base, b.


We’ll call its new point of origin on the left D. We’ll also disconnect line b at point B so that we are now only concerned with a single line instead of a triangle.

Golden Ratio

This line consists of two segments, line a (extending from point D to A) plus line b (between points A and B). We can also think of the new line as an entirety running from points D to B, and, rather than erasing the segment divider in the union of a and b, I will just draw this new line afresh and call it g.

4. The Second Proportion: g to a 

Now, we’ve already brought up proportion of a to b, which we expressed as  x1.

Since line g is the sume of lines a (value: x) and b (value: 1), the length of g is x + 1.

g = x + 1

Since the variable we assigned to line a is x, the proportion of line g to line a is  x + 1x

Larger ratio

5. The Golden Ratio

Now, it turns out that in the Golden Triangle these two proportions (a to b and g to a) come out as equal, and this is the "Golden Ratio." Euclid called it the proportion of the mean to the extreme. Expressed in words, it says that

The proportion of the larger segment (a) to the smaller one (b) 

is equal to

the proportion of the entire line (g) to the larger segment (a).

As soon as we are stating it in its algebraic form, we are doing something that Euclid probably never dreamed of, given the clumsy letter-based system of numbers the ancient Greeks had to work with.

golden ratio

6. Working Towards Finding a Value for φ 

Substituting the one variable (x) and the one value we have (1) we have for the names of the lines (a, b, g), we get:

Golden Ratio

Now that it has served its purpose of displaying its position in setting up the ratio, we can drop that silly 1 on the left side, and we get

golden ratio formula

Let’s eliminate the fraction on the right side by multiplying both sides by x. Then we have:

x2 = x + 1.

We can subtract (x + 1) from both sides [or add –(x + 1), if you want to look at it that way], and we have arrive at:  

x2  – x – 1 = 0.

This kind of equation is called a "quadratic equation," and there are several related ways of solving it. The most famous one is a formula, which is not all that hard to manipulate, but which may strike you as quite intimidating before you've made friends with it. Someone once gave me a mnemonic for it, but I've forgotten it. If anyone know one, please let me know. I wonder if I need a mnemonic for the mnemonic.

Deliberation on the Sideline: Should I bring up the formula here and now? I'm afraid that, regardless of what I write here,  I'll probably never run out of visitors to this blog because there will always be the spammers who like to peddle their wares via the comments sections of my blog, for which I pay to keep it ad-free. A highly prominent subgroup among these stowaways are some gentlemen from Malaysia and Indonesia attempting to lure one into purchasing tours of the world, the luggage you need to travel, and ways of enhancing the companionship you might encounter on such a trip. There are also the ladies from Quebec, fortune tellers one and all, who are eager to use their gifts of clairvoyance on your behalf without asking for anything in return. However, my discussions are not really geared to these friends of free enterprise. I don't know how many people are actually reading any of my streams of conscious with a more serious intent, though I can't bring myself to believe that they could all be spammers. Still, I fear that, based on how it looks at first, if I bring up the formula and plug in its values here and now, it could cut back on my already fragile readership. People who are at home with math probably know more than I do, and don't need to read this. Those who do not could just get scared away by it. Then my Malaysian, Indonesian, and French-Canadian interlopers might just be my only constituency. It's probably best if I leave the formula and its application to a note. Please see note 2 below. 

Skipping over the calculations then, I must tell you now that all quadratic equations have two solutions or "roots," one positive and one negative. Given all of the previous discussion, I'm sure you won't be surprised that the result on computing the ratio is going to result in an irrational number, viz. one with a never-ending string of digits after the decimal points that do not present a predictable pattern. If you use the Wolfram Alpha program, it clarifies for you that the result it gives is only a "decimal approximation"; and while it only carries it to 52 places, but offers to find a longer string should you happen to need it. Livio gives us the first 2,000 digits. His list starts in the middle of page 81 and ends in the middle of page 82. Still, no matter how overwhelmed we may feel looking at this string, an approximation is what it is.

Well, I haven't exactly hidden the fact that the positive number of the ratio derived from the Golden triangle is phi. Here is Wolfram Alpha's "approximation" for the positive root of the equation.


I stated a moment ago, that a quadratic equation comes with both a positive and negative root. Now when we look at the negative root, things start to get weirdly fascinating. The number starts with a 0 rather than a 1 before the decimal point, and it is a negative fraction, but the decimal digits are the same as the ones for the positive solution. 


While we’re at it, let me also mention that there is a number called the "Golden Ratio conjugate," which does not really present us with any surprises. It is defined as φ - 1. This is anything but startling. After all, any number from which we subtract 1 will be 1 less in magnitude. 1.5 - 1 = 0.5. Not worth a letter to the editor or a tweet to the world. 

So, we have

1.61803... - 1,

and its value is quite obviously everything that comes after the numeral 1 and the decimal point. Just for fun (and for visual enhancement, I'll write out the Wolfram approximation: 


Please don't start to yawn just because I mentioned one thing that happened be clear and obvious. Yes, there's no cause for astonishment in the fact that a number minus 1 will be one less than it was before. But the very innocuous nature of this fact highlights the surprise in what follows. 

A bit of nomenclature. Any number used as the denominator of a fraction in which the numerator is 1 is called the "reciprocal" of the number. The reciprocal of 7 is 1/7 , which can also be written as 7-1. We can calculate reciprocals, of course, and for 1/the outcome in decimal notation is 0.14285714..., with that exact sequence repeating indefinitely (and thus making it a rational number). 

Now let's look at the reciprocal of phi, 1/φ, and calculate its value. Here it is, once again availing ourselves of Wolfram's "decimal approximation."


Yes, this is the same number as phi's conjugate. Phi minus 1 yields the same number as 1 over phi. No other number behaves that way. Thus:

φ - 1 = 1/φ

 Let's do one more thing, namely to square phi: The value of φ2  is:  


As you can readily see, it's precisely the same as phi, only larger by exactly 1.

So, these things are true of phi and no other number: 

The negative root of the equation that produces it has the identical decimal points as phi. (-0.618034)
The conjugate of phi is exactly 1 less than phi.
The reciprocal of phi is exactly 1 less than phi, and thus equal to its conjugate.
The square of phi is exactly 1 more than phi.  

Other numbers have their own peculiarities, which may be just as startling, but the traits that you see here belong to phi alone.

Are you beginning to see now why I’m talking about beauty within the numbers themselves? Why numbers sometimes appear to have personalities, at least for those of us who have a little bit of inclination towards fantasy? That a number can be distinct from all others in more than a trivial sense, and that some are more eccentric (or gifted?) than others? That you don’t have to count generations of rabbits or petals of a rose blossom to find fascinating and startling facts about phi (though it may be fun, and we'll get to that)? Other numbers have their own peculiarities, which are just as startling, but the traits that you see here belong to phi alone. The beauty is already there in the number. The rest, which don't want to minimize but just ignore right now, is over and above the properties of the number alone. 

Sorry, we're still not ready to talk about Fibonacci and his series of numbers; we need to stay with its geometrical home for a little longer. In this entry we have stipulated the Golden Triangle, knowing in advance how the ratios and numbers would fall out. The question we need to address is whether this geometric figure is anything other than something concocted by an ancient mathematician for the entertainment of his guests on long November evenings. Can we find the Golden Triangle somewhere where it is right in place, playing a significant role in geometry?

Let's go to the Pentagon to find out!



Note 1: Since we know that both a and c are of a different length than b, we could also use another approach and let x represent the difference between b and either a or c. Then we could say that a and c are of length 1+x, where x could be a negative number or zero, just in case that the "Golden Triangle should turn out to be either obtuse or equilateral. This is not a serious consideration, however, since we already know from the values for the angles that the triangle must be isosceles and acute. So, there's no need to add that complication.

Note 2: This is a quadratic equation, and if you’ve had a bit of math, you may have spent some time puzzling them out. They have the general form

ax2 + bx  + c = 0

The two solutions (or “roots”) can be found with the formula:

Quadratic Formula from SOS Math

(The above image is hosted SOS Math, a very straight-forward and careful website.)

Note the “±" in the numerator. It is this dual operator that helps us find the two solutions, one positive and one negative.

a is the coefficient for x2 ;if there is none, a 1 is implied;
b is the coefficient for x; again, if there is none, a 1 is implied.
c is the modular constant; if there is none, then the value for c is 0.

In our specific case,

x2  – x – 1 = 0.

a=1, b=-1, and c=-1.

Once you go through the two stages of computation for each root (or let Wolfram Alpha do them for you), you arrive at the values for phi, as mentioned above in the text.


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Saturday, November 14th 2015


Militant Islam, part 3

  • STATE OF EXISTENCE: still tired

And now to ISIS. As announced I will not give a history of this organization (except in the sketchiest terms), nor go into all of their present machinations in detail. The Wikipedia site is flush with details and will definitely send a chill down your spine. My goal here is the rather modest one of putting ISIS on the map of Islamic groups.

One very basic truth needs to be reiterated: Islam is never complete as a religion unless it encompasses a state, viz. a political entity. (For al-Qaeda, such a state would be a pure theocracy, arrived at by ungodly measures.) Thus, the fact that ISIS has the establishment of a state as its goal should not be surprising. Still, when my father and I were talking about Islam today on our weekly Skype, we agreed that neither the media nor the vast majority of politicians has yet to catch on to the fact that Islam cannot function as a private “faith,” but must establish the community (al-ummah) as a state. Islam and Christianity are very different in that respect—or should be, and I don’t want to go further along that line for now. See my article “God in the Early Twenty-first Century.”  http://wincorduan.net/God and Ayodhya.pdf

ISIS is not Wahhabi, though there are many similarities. It promotes a supposedly “pure" kind of Islam. It identifies itself as Salafi and stresses the importance of tawhid, the oneness of God and the worship of him alone. Any practice that could be deemed to be shirk (idolatry) must be eliminated. But none of these matters require any link to Wahhabism other than a conceptual one. By engaging in aggressive warfare in order to expand its boundaries and killing Muslims as well as non-Muslims for merely political reasons, it goes far beyond Wahhabi ideology. And, for that matter, so does the genocide of non-Muslims apart from any manufactured war, which includes women and children, utilizing some of the worst tortures ever invented by monsters disguised as humans. Christians and Yezidi are easy targets for them; consequently there is now a growing community of Yezidi refuges in Germany. Please see my post about the Yezidi. http://wincorduan.bravejournal.com/entry/144001

Nor is ISIS Qutbite. The movement did arise out of al-Qaeda with its underlying Qutbite­­ ideology, and it definitely shares Qutbite radicalism in designating all other Muslims to be in a state of jahiliyyah and, thus, making them legitimate targets of aggressive war. But al-Qaeda and ISIS could not coexist for long since ISIS is all about setting up a new government, while Qutbism eschews them all. In the matter of persecuting Muslims, ISIS has emphasized fighting against what they are calling “Shi'ite oppression." This notion would be laughable if it were not accompanied by such inhumane violence and brutality. It is true that since the American involvement in Iraqi politics the Shi’ites have had a greater amount of authority in Iraq than in most of Iraq’s history. However, Shi’ites do represent a majority of Iraq’s population, and, as I stated above, they have lived almost perpetually under Sunni governments. ISIS considers itself as neither Sunni nor Shi’ite, just as was the case for the Kharijites in the seventh century, but their main victims within Islam are the Shi’ites. Outside of Islam, it appears that anyone else is fair game for persecution.

Most significantly, in contrast to both Wahhabism and Qutbism, ISIS has resurrected the position of caliph. Thus, at this moment, they occupy a unique slot. Once again consulting history, after the initial skirmishes, the Umayyad dynasty held the caliphate until it was replaced by the Abbasids in 750. The only territory to which the Umayyads held on was Iberia, and they continued to designate their leaders as caliphs. Thus there was the truly powerful Caliph of Baghdad and the vestigial Caliph of Cordoba. The Abbasids eventually lost power, and in the eleventh century, the strongest caliphate was held by the Shi’ite Fatimid dynasty in Egypt. Thus, there were now three caliphs, the Caliphs of Baghdad and Cordoba—both of them dysfunctional—and the Caliph of Cairo. After the Ottoman Empire had consolidated the Muslim world under its rule, the Sultan also bore the title of “Caliph." The revolution by the “young Turks" in 1922 (that’s where the term originated) ended that practice, and, after several failed attempts to sustain the position in some way both inside and outside of Turkey, since 1924 there has no longer been a caliph of global recognition. I must add here, though, that caliphates in smaller settings are not unheard of, as I shall discuss in a future post. 

ISIS has now attempted to revive the caliphate on a global scale. Its present leader goes by several names and titles, and I will only touch on a few highlights. Born as Ibrahim Awwad Ibrahim Ali al-Badri al-Samarrai, he is the Emir (“prince" or “ruler") of ISIS. His popular name has been Abu Bakr al-Baghdadi, and you will immediately recognize the importance of “Abu Bakr" in that construction. Lately he has added “al Qurashi" to his title, implying descent from the prophet, who, you remember, was of the Quraish tribe. The shortest version with which one can refer to him is “Caliph Ibrahim."

Note: Much of the above is taken from the earlier posts with some modifications. I have not changed the passages below (except maybe for some spelling or punctuation mistakes) because it seems to me that matters have not changed all that much. ISIS has grown over the last year. Has there been progress in putting a halt to it, other than some trophy killings among their leadership?

ISIS displays all of the worst traits associated with the stereotypes of Islam, and in a world where it seems to be impolite to call an evil person "evil," surely this organization and its leader deserve that appellation. The U.S. has taken the lead in attempting to put an end to the travesties of ISIS. I’m not fond of the idea of the U.S. being the policeman of the world, but such a horror cannot be allowed to go on unimpeded. Some European countries have now joined the American effort, and the new Iraqi government is attempting to put an end to ISIS. Perhaps together they will succeed. Also, other Muslim countries have denounced ISIS.

And that last remark once again brings me to a rhetorical question that I have repeated several times. Why are the so-called moderate Muslim nations not playing a more active part in the war against ISIS? For Muslims to sit on their hands while they make frowny faces and verbally dissociate themselves from ISIS is simply not enough. If Islamic countries want to be taken seriously by the outside world, they need to earn that respect by neutralizing those groups that clearly violate the standards of Islam.

Perhaps Western-style democracy is not yet appropriate for some countries where tribalism is too deeply ingrained; simply imposing it seems to backfire in many cases, though I wish it weren’t so. But even an absolute ruler can and should abide by the Qur’an if he is a true Muslim. “There should be no compulsion in religion" (2:256). “People of the Book" (e.g., Christians and Jews) must pay the unbeliever’s tax (jizya) and occupy a lower standing in society than Muslims, but they should be allowed to live and worship God in their own way (9:29). (And, please, there is no way in which one can plausibly rationalize, let alone justify, Caliph Ibrahim’s and ISIS’s actions as a consequence of the Crusades. Unfortunately, if you’ve had conversations with Muslim apologists, you might not be surprised if someone at this very moment is attempting to do just that. )

Muslim leaders, if you want us to see any credibility in your claims concerning Islam, please join actively in the effort to put ISIS out of business!

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Saturday, November 14th 2015


Militant Islam, part 2

  • STATE OF EXISTENCE: just as tired

Continued from the previous post.

The Kharijites

Taking Aïsha out of the picture did not uncomplicate matters very much. The big question between the followers of Ali and Muawiyah was, "Should the caliphate belong 1) to someone who was highly regarded in a particular tribe or b) to a descendant of the prophet?" A third group, called the Kharijites (“Dissenters") emerged with the message that neither criterion was true to Islam. They observed that it had hardly been a bare thirty years after Muhammad’s death, and already the people had lost their way, treating the caliphate as though it were a kind of monarchy. They insisted that the truest and purest of all Muslims should be the one chosen to be caliph, even if he had been nothing more than a slave boy. Descent or social standing should have nothing to do with the selection. The fact that such inappropriate criteria were being used to designate the caliph, led the Kharijites to conclude that many supposed Muslims had become apostate. They had already fallen back into the ignorance and darkness of the time before Muhammad (the jahiliyyah). If so, these lapsed Muslims were considered to be worse than unbelievers and potentially subject to execution. K

The Kharijites acted on their beliefs. When Ali started to negotiate with Muawiya rather than fight him, a group of Kharijites assassinated him. They did not think of themselves as either Sunni or Shi’ite (which were still in their embryonic states).

While we are touching on the Shi’ites, it is important to realize for what follows that Shi’ite Islam has constituted the majority population of both Iran and Iraq once they became Islamic. However, the area that we now call Iraq has been governed by Sunnis for almost all of that time: the Umayyad dynasty followed by the Abbasids, the Selkuk Turks, the Ottoman Turks, the Hashemite monarchy (imposed by Great Britain), and the B’ath party of Saddam Hussein.­­­

As a distinct group the Kharijites did not last very long, but similar movements, which we can call “neo-Kharijite” have popped up again and again during the history of Islam. Please read my website, Groups of Islam, or Neighboring Faiths for more of what transpired as I will try to limit myself to only the most relevant parts here. The point is that these neo-Kharijite groups would not have arisen if Islam had been practiced in the pure, unadulterated form that they expected. Some developments within Islam went into the direction of mysticism (i.e. Sufism), as well as adaptations to folk religion and superstitions. 


One response to these alleged deviations came from Muhammad ibn Abd al Wahhab (1703-1792), who initiated a significant reform movement in what is now known as Saudi Arabia. When most of the Arabian Peninsula came under Saudi rule in the early twentieth century, “Wahhabism" became mandatory, and anyone who did not comply with its teachings was liable to be executed. It is a rather rigid form of Islam, forbidding anything that could be interpreted as idolatry (shirk), such as the veneration of Muslim saints or their gravesites, not even making an exception for the burial site of Muhammad in Medina. A central concept in Wahhabism is tawhid, which basically means “oneness” and refers to the one God (Allah) alone as being worthy of worship.

The Wahhabi movement came about in order to purify Islam. Countries that adopted it have not been hospitable to other religions or versions of Islam, but on the whole, their initiatives have been internal. In addition to Saudi Arabia, the two other countries in which it has made its home are the United Arab Emirates and Afghanistan, where the Taliban forced Wahhabi interpretations of the Qur’an on the people for a time (and would like to do so again). But there is no caliphate in Wahhabism. The kings of Saudi Arabia claim to have a divine mandate to rule, but they do not refer to themselves as caliphs.

One of the crucial tenets of Wahhabism is that no Muslim should follow human teachers in their beliefs and practices. Consequently, Wahhabis do not care for that label and prefer to be called Salafis, which means that they follow the pattern set by Islam under the first three caliphs.  We will meet the term Salafi again. Whether they like labels or not, Wahhabism is clearly a part of Sunni Islam, and, if I may give you some terms without explaining them here, they are Ash’arite in outlook and Hanbalite in their shari’a. 

Qutbism and al-Qaeda

Even though Osama bin Laden (1957-2011), the eventual leader of al-Qaeda, grew up in Saudi Arabia under Wahhabi teaching, he eventually turned into a different, far more radical direction, inspired by the writings of Seyyid Qutb (1906-1966). Qutbism and its philosophical allies deny the legitimacy of any human government (including a caliphate) and look for a totally Islamic world, which will be governed by Shari’a alone. According to them, not only non-Islamic countries, but also all Muslim countries that are governed by human beings (and they all are, without exception) are in the state of jahiliyyah (darkness and ignorance). The Saudi government is not only included in that judgment, but was singled out by Osama over and over again as a case in point of such apostasy.

I’m sorry for playing the same record so many times over the last few years, but I must continue to urge you to read Milestones by Seyyid Qutb. I’m certainly happy if you just take my word for the summary, but if there is any doubt in your mind concerning the utter destructiveness of some Islamic ideologies, this book should resolve it for you.

The Qur’an teaches that Islam should not be propagated by the sword (Sura 2:256). People should be able to come to a free decision of the truth. Consequently, many Muslim apologists today go to extraordinary lengths to explain away the various aggressive wars fought by Muslims ever since its birth. For our purposes right now, this issue is irrelevant because al-Qaeda and other followers of Seyyid Qutb not only acknowledge that there have been Muslim wars of aggression, but advocate that they must be resumed.

Qutbites agree that one should only become a Muslim by free choice when one recognizes the truth, but declare that people today do not have true freedom to make such a choice since they are “enslaved" to human governments. Thus, the primary item of their present agenda is to abolish governments, both those of non-Muslims and of the multitude of pseudo-Muslims who live within the recrudescence of jahiliyyah. In order to make a truly free choice concerning Islam, people need to live in a truly Islamic environment, and that is one in which only Shari’a is needed without people to enforce it. In short, it takes global violence in order to arrive at a state where only one ideology (Islam) rules, and then, finally, people will be free to choose whether they want to be Muslims or not.

Let me repeat a point that I’ve made several times in various places in order to illustrate this idea. It has been stated multiple times that al-Qaeda’s attack on the Twin Towers on 9/11/2000 was irrational because it not only killed non-Muslims, but Muslims as well. This particular inconsistency vanishes under the Qutbite paradigm because any supposed Muslim working in the Twin Towers would surely be in the state of jahiliyyah, and, thus, be subject to destruction just as much as any other infidel. Anyone who is not a Muslim in accord with the principles of Qutb and al-Qaeda is a hypocrite, and the Qur’an destines hypocrites to a worse fate than unbelievers. Sura 4:145: “The Hypocrites will be in the lowest depths of the Fire: no helper will you find for them; -“

Next installment: ISIS

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Saturday, November 14th 2015


Militant Islam, part 1


I’m posting this entry, which is a patchwork quilt of some earlier ones, in response to the bombings carried out by ISIS members in Paris on 11/13/2015. You cannot fight the enemy that you do not know. There are three parts, basically ready for upload, so they should all be up within an hour or so, after which I may revise a sentence or two and add a few pictures. 

Current events have created the need to interrupt my series on phi, and I suspect, if it were not for the reason why I’m doing so, there could just be a few sighs of relief from my loyal readers. Disloyal ones, if I had any, would not even have tried to get through what I have posted along that vein. But I will get back to math shortly; in fact, I have an entry all written out, but this topic should take priority.

Yesterday (Friday) afternoon I once again visited the class on world religions at Taylor University, which is taught by Dr. Kevin Diller. As I did last year, I gave an overview of radical Islam, beginning with the clashes about who would become Muhammad’s successor and ending up with the Taliban, Al-Qaeda, and ISIS. I also once again tried to include the point of view of Aïsha, one of Muhammad’s widows, who exercised a great amount of influence on the events following Muhammad’s death and the formation of the Hadith, the narrative supplement to the Qur’an. Furthermore, it is impossible to understand Aïsha’s life and actions and gloss over the violence and hate that characterized already the very first generation of Islam’s existence. How sad that, just a few hours later, as we turned on the news, we got an audio-visual lesson on militant Islam and the terrorism it engenders!

In this entry I’m combining much of what I posted last year as a background for being aware of ISIS with the material of the Aïsha lecture. ISIS was not yet in existence when I wrote the second edition of  Neighboring Faiths, and last year’s entry on Aïsha, which has more details than the book, is no longer accessible due to an apparent incompatibility of my code with Bravenet’s preferences. The current edition of Neighboring Faiths has two chapters on Islam, the second one being devoted entirely to the radical movements. So, if you would like further information on what I’m writing about here, you’ll find a lot of it—with painstaking documentation—in chapter 5. Also, please visit my lengthy website “Groups of Islam” for more background and some details that I cannot go into now. 

Please keep in mind that most complex scripture-based religions expect to be the only religion at the end of time; therefore, global ambitions per se should not be considered unusual or the mark of an invidious religion. However, there are important differences in the means of getting there. Spreading a faith by works of mercy, verbal proselytization, or relying on the direct action of God is not the same as conquering the planet for your religion by physical warfare. If you’re familiar with my previous writings on the subject, you already know that, various protestations by Muslim apologists notwithstanding, some Islamic groups subscribe to the latter method—and do so openly and without expressions of regret, let alone apology. They include both al-Qaeda and ISIS.

The biggest division within Islam is, of course, the split between Sunna and Shi’a, and it began right after Muhammad’s death. Who among those who had been his Companions (an honorific title as well as a descriptive noun) would lead the new community (al-ummah) established by the prophet? Muhammad’s son-in-law, Ali ibn Talib, not only seemed to have the inside edge, but also claimed to have received Muhammad’s designation and his spiritual powers. Nevertheless, the consensus (Sunna) went with Abu Bakr, Muhammad’s close friends and the father of Aïsha, his all-too-young and spirited wife. The rulers of the Islamic world became known as the “caliphs." Thus we have the majority, the Sunna, represented by Abu Bakr, and the “dividing party," the Shi’a, associated with Ali. The first three caliphs, Abu Bakr, Umar, and Uthman were acclaimed by the Sunna. Ali finally became the fourth caliph, and theoretically the breach might have been healed, but by that time the divisions within the Sunna and between Sunna and Shi’a were already unalterably headed towards mutual bloodshed.

Aïsha was not just sitting on the sidelines watching the events unfold, but took an active part in them. I think it’s safe to assume that among the majority of Westerners who are aware of her at all, the extent of their knowledge ends with the fact that Aïsha was betrothed to Muhammad at a very young age, and the marriage was consummated when she was still far too young for such things (six and nine years old respectively are the most commonly cited ages). Child marriage and consummation prior to puberty were a dark practice in a dark culture. Still, to understand the early history of Islam, and thereby Islam itself, after we have exercised our moral judgment, we need to look past our repugnance and learn from the facts as they were reported.

Aïsha was Muhammad’s favorite wife for the last nine years of his life, and she supported him publicly throughout that time. However, she had a head of her own right from the start and was not above criticizing the prophet himself in private, e.g., for his claims to have had revelations that allowed him to slip out of inconvenient situations. When Muhammad was on his death bed, the other wives conceded that he could stay in her quarters because they all knew that Muhammad and Aïsha had a special relationship, something that had been missing from his life during the most turbulent years of his career, which occurred (not without connection) after the death of his first wife, Khadija, his only wife as long as she lived. (There is actually no certainty on how many wives Muhammad had later on; 10 or 12 are popular options.)

After Muhammad’s death, Aïsha was very concerned that Islam should stay true to the prophet’s teachings. Because of her close relationship to him, she frequently served as consultant on the manner in which Muhammad himself had practiced what he had taught. Consequently, Aïsha became a public person, who also got involved in the ensuing political matters, including the issue mentioned above of who should be caliph.

The honor of being the first caliph went to Aïsha’s father, Abu Bakr, so things went for her as she had hoped they would. She also approved of the second caliph, Umar (another father-in-law to Muhammad), but the political aspect of the Islamic world was heading towards great turmoil. Umar was assassinated, and the next caliph, Uthman, perhaps best known for collecting the authoritative edition of the Qur’an, was not as capable a leader as his two predecessors.

An important factor that came into play was the caliph’s clan (subtribe) affiliation. Uthman came from a strong clan within the Quraish tribe, the Umayyads, while Muhammad hailed from its Hashemite segment. The Umayyads had been Muhammad’s strongest opponents and the very people who had caused the prophet to flee from Mecca to Medina (the hijra). After Muhammad had conquered Mecca, they were the last among the clans to convert to Islam.  Aïsha was not alone in thinking that a member of the Umayyad clan should not be caliph. Furthermore, it appears that Uthman’s interpersonal and leadership skills were not adequate for the job.

Aïsha thoroughly disliked Uthman. However, it is one thing to oppose the ruling caliph personally and politically, it is quite another to humiliate him and kill him and his family. The new governor of Egypt, Abdullah ibn Saad, did exactly that with the aid of a thousand or so rebels, perhaps motivated by his own ambitions toward the caliphate. Aïsha was outraged. She entertained a strong hope that the next caliph would take stern measures to avenge Uthman’s death.

The next caliph turned out to be none other than Ali ibn Talid, which made the matter all the more awkward from Aïsha’s perspective. She did not care all that much for Ali either. For one thing, he had been seeking the caliphate in competition with her father (Abu Bakr); for another, his spine was just weak enough that he could be manipulated into changing his mind on crucial issues, sometimes as though it were on a whim at the very last moment before some action should have been taken. Ali did not prosecute those who had abused Uthman, perhaps because it was not possible to identify the specific people involved; they had found refuge in the anonymity of being soldiers in Ali’s army. For Aïsha, Ali’s inaction in this case was another sign that he was no more suited for the position of caliph than Uthman had been.

There was another complication. In this, the fourth, round of choosing the caliph, Ali still had competition. It came from a man named Muawiyah, another member of the Umayyad clan, who thought that he was entitled to the Caliphate primarily because of his tribal status. So, both Ali and Muawiyah were odious to Aïsha, but she was particularly angry at Ali for letting the mistreatment of one of Muhammad’s Companions (i.e. Uthman) go unpunished.

Aïsha, now around thirty years old, gathered an army and ransacked the town of Basra, around which Ali’s troops had set up camp. Then she went on to confront Ali in pitched battle. She personally served as commander of her troops, perching on top of a camel behind the lines so that she could have a clear view of the field and give orders as necessary. Thus, this encounter has been known as the “Battle of the Camel.” Sadly for Aïsha, Ali’s troops were stronger than hers, and she lost the battle. Ali’s men captured her, and Ali had her escorted back to Mecca. She remained there for a short while, and then moved back to her original home in Medina. Her days of political involvement were over, but she taught the message of the prophet and became one of the most important sources for the secondary writings concerning Muhammad, known as the hadith. (In case you were wondering earlier, that's how we know about some of her private conversations with her husband.)

Next installment: The First Radical Muslims and their Offspring

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Tuesday, November 10th 2015


Pythagoras, Music, and Rational Numbers


First of all, here is a diagram of the different kinds of numbers: ℕ    ℤ    ℚ    ℝ    ℂ, which I discussed in the last entry.

Types of Numbers

I left off by saying that the number that is the subject of our attention, namely φ (= 1.61803…) belongs to the category of the “irrational numbers,” which, together with counting numbers (ℕ), integers (ℤ) and the rational numbers (ℚ) make up the set of so-called real numbers (ℝ). These irrational numbers received that label because they did not fit into the way in which the Early Greek mathematicians, particularly Pythagoras, attempted to envision the numerical regularity of the universe.

Pythagoras is known, among many other things, for his discovery of the linkage between numbers and music. Some people hold it against him that he supposedly mechanized the beauty of music by imposing his mathematical system on it. However, Pythagoras did nothing of the kind. He did not impose anything; he discovered that the sounds that we consider to be pleasing follow some simple numerical patterns. The music came first; Pythagoras found the numbers that appeared to be hidden in it.

I tried on a couple of recent shows on StreetJelly to demonstrate just a tiny bit of what had Pythagoras so fascinated, but it probably didn’t have as much entertainment in that context as I had hoped. So, I made a video:

Pythagoras and his followers were rather pleased with the notion that the universe could be described with ratios of whole numbers, the “rational” numbers. Now, somewhere, somehow, someone discovered that it was not so; there were numbers for which it was impossible to represent them accurately as whole-number ratios. There are many legends connected to the intrusion of irrational numbers into the beautiful harmony of the Pythagorean system. Some of them blame one of his followers named Hippasos and go so far that Pythagoras ordered his execution and had his minions swear an oath that they should never ever reveal the existence of this alleged heresy. The most we can document with reasonable assurance is that, somewhere around the time when Pythagoras flourished, someone discovered irrational numbers, and Pythagoras presumably did not much care for this idea.



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Tuesday, November 3rd 2015


Kinds of Numbers


Last time I tried to show you that the number zero, when put under a little bit of inspection, becomes a whole lot more than just a symbol for "nothing." My intention was to show that it is not just trivially true that each number has unique properties. The number that concerns us in this series of blog entries is 1.61803..., and it is one of the true stand-outs among all the numbers with which we are familiar. In popular literature it has acquired the Greek letter phi, φ, as its moniker, though academic mathematicians prefer the letter tau, τ. There are many resemblances between φ and its cousin, the even more famous pi, π.

Mathematicians recognize several classes into which various numbers fall. These groups can be visualized as concentric circles, beginning with a relatively small one and increasing in size, so that the next larger group includes all of the previous ones, but also adds many more. They are represented by hollowed-out capital letters, which are derived from their German names: ℕ, ℤ, ℚ, ℝ, and ℂ.

  is the set of all natural numbers, the ones we use to count. It does not include fractions or negative numbers. They are the whole numbers that begin with 1 and stretch to infinity. "Infinity" should not be construed as a number; the term simply designates an unending chain of numbers, and it should also be understood separately from the metaphysical concept of infinity when we apply it to God. There are two kinds of numbers among the multitude in the  circle: those that are prime and those that are not. Prime numbers are those numbers that cannot be divided without remainder by anything other than1 (which is trivial) and themselves (which is given). Any prime number divided by an integer other than 1 or itself is going to leave a fraction. For example 57 divided by 1 is 57 and divided by itself is 1, but if you try to divide it by any other whole number, you get a fraction.

61/2=30.5     61/3=20.333...     61/4=15.25     61/5=12.2    etc.

(Please note the difference between the results after division by 2, 3, and 4 in contrast to division by 3. We'll come back to that little feature shortly.)

Prime numbers have given rise to numerous interesting insights, guesses, and speculations. Among them are:   

a) the idea that there is no largest prime number (proven by Euclid around 300 BC);

b) the "Weaker" Goldbach Conjecture, which says that every odd number greater than five can be expressed as the sum of three primes, or, alternatively that every odd number greater than seven can be expressed as the sum of three odd primes (proven definitively in 2013 by Harald Andrés Helfgott, now at Göttingen);

c) the full Goldbach Conjecture, according to which every even number larger than four can be expressed as the sum of two prime numbers (so far unproven);

d) the famous Riemann Hypothesis, which is concerned with the distribution and density of prime numbers* (so far unproven).  

Serious mathematicians have been joined by seekers of fame and fortune, not to mention crackpots, to try to prove c) and d), so far probably unsuccessfully. People who are aware of their limits, as exemplified by your bloggist, may know enough to understand the problems, but are also too smart to even imagine that they could come close to a proof.

  (derived from the German word Zahl), is the next category, and it consists of all integers. It includes negative numbers, and, if there is any question of whether zero belongs into  or not, it is definitely at home within ℤ.

It might surprise you, as it did me, to know that negative numbers are a relatively recent addition to what mathematicians on the whole accept as legitimate. As late as the seventeenth century well-known mathematicians, such as Blaise Pascal dismissed them as "nonsense."

Rene Descartes approved of negative numbers only as long as they could be transformed into positive ones. Thus, if we are looking for the square roots of 1, we would say that there are two correct solutions: 1 and -1, but Descartes wanted to strike the -1, because in its unaltered state -1 is not a true number. However, he allowed the use of negative numbers in intermediate stages that will arrive at a positive number, e.g., -4 x -7 = 28.

Even in the eighteenth century, Gottfried Leibniz, the inventor of calculus as we know it, was hesitant about the use of negative numbers. Leonhard Euler (pronounced "Oiler," not "Yooler), one of the top ten mathematicians of all times, maybe even one of the top five, still felt the need to justify their use by drawing an analogy representing them as a kind of "debt."  Thus, even though  is situated logically between and the next category , historically  was accepted universally only about two thousand years after .  

 The letter is based on the word Quotient, which is also an English word, but is used in German much more commonly for the result of a division problem, similarly to the ease with which we use "sum" and "product" for addition and multiplication respectively. We call its occupants the "rational" numbers. This class contains all of the numbers that can be expressed as fractions without any unspecifiable remainders hanging around. Clearly any member of  is included in , since any whole number can be expressed as a fraction.

E.g., 17 is equal to or example, 17/1 or--perhaps a little more meaningfully--51/3.

Negative integers, who lend their distinctive flavor to belong into , though we must be careful with our signs. -17 is not equal to -51/-3 (that would come out as the positive 17). To maintain the negative sign we need to make sure that the numerator and denominator are of opposite signs (one positive, one negative). What we are adding in  is all of the neat fractions that, if divided out, eventually come to an end.  1/2, 8/25, and 256/3 are rational numbers.

So, 1/2 written in decimal form is 0.5.

8 divided by 25 yields 3.125. The remainder is 3 places long, but it comes to an end.

If we divide out the fraction 58/3, our result will never come to rest. The solution is 19.3333...., and the threes will continue interminably. But it still belongs into ℚ.  For one thing, that awkward number is still considered to be rational since it began its life as the ratio of two whole numbers, 58 and 3. For another, even though the 3s after the decimal point will go on forever, we know that they will never change. There will be 3s and nothing but 3s.

The next group is , where  stands for "real." Resist the temptation to think of  as standing for "rational." The rational numbers are already accommodated by . So, what  brings to the party is all of the irrational numbers, of which there are more than rational ones, according to Georg Cantor and modern set theory. Now, you may think of mathematics as the ultimate exercise in rationality and be somewhat befuddled as to how there could be "irrational" numbers. Well, these numbers are notirrational in the sense that they are contradictory or derived from some mystical intuition. They are "irrational" in the sense that they do not fit into the nice and tidy collection of numbers in  that can be expressed as a fraction of two integers or that, as a decimal, ever reaches a conclusion. To clarify the second point, an irrational number's decimal fractions neither come to an end point nor assume a pattern, such as the repetition of 3's we saw earlier, or even the repetition of an identical set of several numbers.  

Finally, there is , which adds the so-called imaginary numbers, or, much better, complex numbers (hence its letter designation). They do not play a role in our discussion (as far I can see now), but I'm bringing them up because it doesn't make any sense to list all of the others and leave out this last one. Above I mentioned in passing that the number 1 has two square roots, namely 1 and -1. Each of those two numbers multiplied by itself yields 1. So what would be the square root of -1? It obviously cannot be either -1 or 1 because they give us 1, not -1. I guess one could try to sneak in the notion that a square root could consist of one positive and one negative number, but doing so would fly in the face of what we mean by "square root."  So, let's just stipulate that there is such a number, and we'll call it i, for "imaginary," not "irrational," since the irrational numbers were already allocated their place in ℝ.  Philosophically, one can debate whether i is any more or less imaginary than, say, -243; some folks might say that they are equally imaginary. Others, among them your bloggist, would endow them with the same reality as all other numbers. Regardless, with i having achieved acceptance, the infinite set of real numbers (i.e. ℝ) is expanded even further. And, not-so by the way, since their first appearance in mathematics, "imaginary" numbers have also become useful in physics. They are not just applicable when they have no applicability. 

It is possible, even desirable, to combine real and imaginary numbers. But you can only do so with the same style that you would apply to a polynomial, viz. to state them as a real number plus so many i, e.g., 3+2i or 37+8i. The number "line" has become a "field" if we want to have any hope of graphing these numbers. These combinations of real and imaginary numbers gives us them the appropriate label of "complex" numbers.  As with all previous classes, includes all of the numbers enclosed by its circle, in this case the real numbers and all of their subdivisions. The numbers that do not have a need a direct reference to i can all be rewritten as "n+0i."   

Let me get back to phi now with a final comment for tonight. This number, which we shall treat in its own right eventually, does not appear in our concentric circles until we get to , the real numbers. It is neither a counting number, nor an integer in the larger sense, nor a rational number.  As we described above, that means that any expression of it as a proper fraction is only an approximation, and, for that matter, any expression of it as a decimal number is also only an approximation, though it has been calculated to thousands of places. This is one of the ways in which phi and pi resemble each other. 


Today I had my second PT, and I feel as though a truck had barreled over me (insofar as I can imagine such a thing since I've never had a truck roll over me in reality, not even a mini-pickup). We are having an unusually warm fall so far, accompanied by very little rain, so it's been nice to be outside. I wish it would last this way for many more weeks or even months, but I'm pretty sure what the realities will be.    

*The actual formulation of the Riemann hypothesis is rather opaque unless you've studied it just a bit. “All non-trivial zeroes of the zeta function have real part one-half.”  

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Sunday, November 1st 2015


Numbers: Zero and her Friends

  • STATE OF EXISTENCE: Schlepping Along

So, did you do your additions and divisions as suggested on the last post? Did you come up with that strange number 1.61803 ...?

To take things one step further, are you aware of the fact that you were implementing a slightly altered version of a recursive function that yields the so-called Fibbonaci numbers? (Don't worry if you haven't; I'm intending to explain this thing to you in this series.) Going one step further, have you been taught that the proportion of 1:1.618 is derived from this series, and that it is often called the "Golden Ratio"?  Finally, are you familiar with the idea that the "Golden Ratio" is supposed to evoke our deepest aesthetic sensitivity for what we label as beautiful?

If so, thanks for playing along. As for me, I have a lot of work cut out for me now because the concepts I referred to above embody some half-truths. I have seen them applied multiple times in the area of Christian apologetics and, thus, should be corrected, lest we embarrass ourselves (not that I think that what I write here will have a widespread effect). It's going to take me a little time to bring up all of the necessary background so that I can make my point without simply throwing one dogmatic assertion against another one, as Hegel might say. Needless to say, doing so means distributing my discussion over a number of entries, particularly since I want to commit myself to keeping any single entry from becoming too large. (I know, "promises, promises ... ")

Livio Golden RatioThe book to which I made reference in the last entry is Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (New York: Broadway Books, 2002). I will return to it for a lot of detailed information in a while.

Before going on, let me clarify that I'm going to lapse into anthropomorphisms from time to time, talking about numbers as though they were personal causal agents. Please understand that doing so is pure whimsy on my part, and that I do not ascribe any powers to numbers beyond their mathematical contexts. For example, I don't do Gematria or think of numbers as lucky or unlucky. The latter concern, it seems to me, becomes impossible because different cultures have differing numbering systems and opinions about numbers. A number that is considered auspicious in one culture, may be thought of as potentially harmful in another one, provided that it is even expressible there, and there can be no standard to measure the truth of something that's supposed to be beyond reason and logic. 

Over the last dozen years or so, I have become increasingly fascinated by the beauty of numbers, which I believe to be God's creations. Each one has its own properties; no two of them are alike. --- Okay, I know there's not a whole lot of profundity in that observation as it stands. Come to think of it, there isn't any. --- However, I'm thinking beyond mere tautologies, e.g., not just that 1 is not 2, etc.  I'm also including the special attributes that come with each number, which you may not pick up on until you have spent some time in their company. For example, look at all the things that our friend Zero can do (and, for that matter, what she's not suited for).  

Smiling ZeroWe will definitely miss out on Zero's charm if we merely think of her as "nothing." It is said that some of the early Greeks would not recognize a concept such as Zero because doing so would imply treating nothing as something.

[E.g., Charles Seife, Zero: The Biography of a Dangerous Idea (New York: Penguin, 2000) BTW, this book has some good points, but is rife with many common false generalizations concerning Greek philosophy, the Middle Ages, Aristotle's standing in the Middle Ages, the Church, and whatever else continues to play to pseudo-intellectual audiences].

Alas, the grammars of IndoEuropean languages with which I am familiar, make it unavoidable to speak of "nothing" as though it were an object. Still, even though it doesn't make sense to say that "non-being exists" (i.e. that it has being), that fact doesn't imply that it has no reality. It may not exist in the same way as pumpkins or spiders do, but it can certainly be real. If you open your refrigerator in order to get some food, and it's empty, you're confronted by something real, even though what is real is the absence (or privation) of food. So, it is certainly appropriate that in the process of counting we usually record the absence of something by labeling it with a "0." But that's only one attribute of Zero.

Zero is a place holder in our system of numbers. I've mentioned different systems before, and I'll just refer you to a collection of some of those entries. The point at hand is that in our Indian/Arabic numbering system there's a big difference between the numbers 54 and 504. In the case of the second number, the presence of Zero between the ciphers 5 and 4 has added 450 units to the number. She's telling us that the column of 10s is not represented in the numeral, which is a far cry from thinking of her as just "nothing."  

Moving on, we call on Zero to express "additive identity."

12 + 0 = 12.

We're adding Zero to 12 and still have the same old 12.

In fact, Zero does this for every number. It assures us that, if added to any number, the number remains identical to what it was before the addition. --- "Big deal," you say. "That's just playing word games." --- "Not quite," I must reply. Think of how this property works out in, say, an algebraic equation.

Let's say that you're multiplying two polynomials, e.g., (x+10) and (x-10).

Their product is  x2+10x-10x-100.

Furthermore, we know that 10x -10x=0.

Now we're definitely not fussing around with "nothing" when we recognize that the presence of 0 will not affect the identity of the rest of the formula, allowing us to eliminate Zero safely, viz. without having to maintain either "10x-10x" or "0" in our thoughts or on our papers. If you want to get to know the real Zero, in the long run it will be better to think of her as representing "additive identity" rather than as "nothing."

Queen ZeroIn the kingdom of multiplication, Queen Zero rules! Wherever she appears in a product, she takes total control, regardless of how many other factors are present or how large they are. The number of hydrogen molecules in the universe is equal to the number of moons circling the earth, once they have both been multiplied by Her Majesty Queen Zero.

On the other hand, in the adjoining land of division, Zero plays the part of an obstreperous brat. Either we give her everything (which we are not about to do), or she won't play at all. One could make use of the sleight of hand that mathematicians call "approaching the limit" to argue that any number divided by zero yields infinity, and I must state for the record that a feat along such lines is fundamental to the theory behind calculus. However, in the usual arithmetical sense, dividing by zero is not just something that you should not do, but something that you cannot do. I mean, can one even conceive of what dividing by zero means?

One more unique aspect of Zero's personality is her wanderlust. She comes and goes within the community of numbers depending on the circumstances. If you draw a line of integers moving from say -5 to 5, you will find her situated comfortably between -1 and 1. But that's only for "counting numbers," viz. cardinal numbers. For example, if we want to count the number of moose in my backyard, we'll start with 0, and--under normal circumstances--stop there as well. If we were to drive to Yellowstone Park and count how many moose we have seen while there, we'll have to start with 0, but hopefully move up from there along the ladder of integers. However, if we leave the cardinal numbers and step over into the realm of the ordinal numbers, we find that Zero is no longer with us. We can talk about the 1st moose we saw, perhaps followed by the 2nd, 3rd, 4th, etc., but it doesn't make any sense to say that before we started out we saw the 0th moose.  (The same thing applies to bears, elk, geysers, tourists, and forest rangers.)

What's more, Zero is also absent in slightly hidden applications of the concept of ordinal numbers. Take, for example, our division of history into two segments, usually designated as BC (Before Christ) and AD (Anno Domini -- the year of our Lord) or some supposedly neutral labels, such as BCE ("Before Common Era") and CE ("Common Era"). We need not occupy ourselves with the question of whether the calculations that went into setting up this system were accurate. The point is that, whatever year was thought of as the year of Christ's birth became the year 1 AD (or AD 1 for purists). It's predecessor was "Year 1 BC." There was no "Year 0," and, contrary to Seife who sees this fact as a part of the intellectual impoverishment of the Church, is how it needs to be. 

I'll stop here with my observations on Zero. We could go on in the same way, sliding up to 1, and we would find out some really entertaining properties for that number. The same goes for 2, as well as for 3, 4, 5, and really "everyone" else present at our little party for numbers. But we'll postpone such ruminations for now and try to get back to 1.61803... Here is a number that's not only unique in an obvious sense, but that's downright eccentric.



Yesterday was Halloween (as well as Reformation Day). Once again my puppets got to sit on the front porch as decoration for the festivities, and the children really liked them. For the most part, they're "monsters" in the generic sense, but friendly-looking. I am, of course, referring to the puppets, not the children, though they were also friendly-looking, even in their disguises, and they were exceptionally polite. It's always fun to watch the older ones, say, seven years old, look out for their younger siblings, who might be four or five. All but the oldest kids (like high school age) were accompanied by adults, some of whom stayed in the background, though some also came to the door. Yes, some accompanying parents even had their own collecting bags, and did not mind accepting a Snickers or Kit-Kat. A lot of the trick-or-treaters came in motor vehicles, obviously car-pooling from out-of-town.

The official time for trick-or-treating here in Alexandria was 6 pm to 9 pm. By 5:55, the first installment of about twelve kids of various ages had already found their way to our front porch. Most of them were girls in princess costumes, but I couldn't really keep track because I needed to focus on making sure that everyone got their share of goodies without my dropping either of the two big bowls in which they resided. Two lollipops and one candy bar was this year's formula. Among the various visitors there was one adult couple who was taking their sleeping baby around; the infant couldn't have been a year old yet. They looked like they were just having a lot of fun. Then there was a young man, in his early twenties I would guess, also carrying a sleeping child who appeared to me to be just a few months older than the previous one. Somehow I got the feeling that there was a back story that involved a little bit (or maybe a lot) of hurt, and June picked up the same vibrations.  All in all, we had sixty or more callers, measuring by the amount of candy that was left by 8:30 when things had become quiet. I turned off the porchlight and brought the puppets back in, thankful for the parents that cared for the safety of their children as well as for the apparent fact that good old Smalltown USA is still a place where kids can go trick-or-treating in safety. And I hope that our house will continue to be known as child-friendly in future years.


On Thursday (10/29), I finally had my first physical therapy session after my mini-stroke, which had occurred on 9/8. It's going to be challenging, but I'm sure it will be helpful. The goal is to restore some strength to my left side and my sense of balance. Neither one has been right for many years now, but the incident aggravated the problems, and I hope we can maybe get back to pre-TIA conditions.

Oh, yeah, I should probably mention that June got attacked by a fair-sized alligator a little while ago, as you can see in the picture, but she's doing fine.

June with Alligator

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